bounded functions $g(\mathbf{w})>0$ and $\beta(\mathbf{w})$. That is, we have

$\delta_{L}(r,\mathbf{w})\sim
g(\mathbf{w})r^{\beta(\mathbf{w})}\exp(-r(\lambda(\mathbf{w})-1)),\quad
r\to\infty,$ (25)

The independence copula is a trivial example, with $g(\mathbf{w})=1$ and
$\beta(\mathbf{w})=0$. Similar forms were also considered by [36] for
exponential margins.

If $\delta_{L}(r,\mathbf{w})$ has asymptotic form (25) with
$\beta(\mathbf{w})=0$, then
$f_{R|\mathbf{W}}(r|\mathbf{w})\sim\tilde{g}(\mathbf{w})r^{d-1}\exp(-r\lambda(\mathbf{w}))$
for $r\to\infty$, where
$\tilde{g}(\mathbf{w})=2^{-d}(g(\mathbf{w})/f_{\mathbf{W}}(\mathbf{w}))$. In
this case it is more accurate to apply the assumption of a GP-type
(exponential) tail to $\delta_{L}(r,\mathbf{w})$, rather than to the
conditional radial density. This motivates an alternative POT assumption for
the conditional radial density on Laplace margins, that

$f_{R|\mathbf{W}}(r|\mathbf{w})\sim\frac{r^{d-1}}{A(\mathbf{w})}\,\exp\left(-\frac{r}{\sigma(\mathbf{w})}\right),\quad
r\to\infty,$ (26)

where $\sigma(\mathbf{w})=1/\lambda(\mathbf{w})$ and $A(\mathbf{w})$ is a
normalisation constant. The RHS of (26) is a gamma density function with shape
$d$ and scale $\sigma(\mathbf{w})$, and hence
$A(\mathbf{w})=(d-1)!\,(\sigma(\mathbf{w}))^{d}$. The difference between the
assumption that the tail of $f_{R|\mathbf{W}}$ is exponential and the
assumption that it is gamma, becomes larger as the number of dimensions $d$
increases. Therefore, in many cases assumption (26) will be a more accurate
basis for SPAR models on Laplace margins than model (4), where the tail of
$f_{R|\mathbf{W}}$ is approximated with a GP distribution. If assumption (26)
is assumed to hold above some threshold $\mu(\mathbf{w})$ with corresponding
exceedance probability $\zeta(\mathbf{w})$, then a modified version of model
(4) for Laplace margins can be written as

$f_{R,\mathbf{W}}(r,\mathbf{w})=\frac{\zeta(\mathbf{w})f_{\mathbf{W}}(\mathbf{w})}{A(\mathbf{w})}r^{d-1}\,\exp\left(-\frac{r}{\sigma(\mathbf{w})}\right),\quad
r\geq\mu(\mathbf{w}),$ (27)

where the normalisation constant becomes

$\displaystyle A(\mathbf{w})$
$\displaystyle=\int_{\mu(\mathbf{w})}^{\infty}r^{d-1}\,\exp\left(-\frac{r}{\sigma(\mathbf{w})}\right)dr=(\sigma(\mathbf{w}))^{d}\Gamma\left(d,\frac{\mu(\mathbf{w})}{\sigma(\mathbf{w})}\right),$

and $\Gamma(\cdot,\cdot)$ is the upper incomplete gamma function. In the case
$d=2$ we have

$\displaystyle A(\mathbf{w})$
$\displaystyle=\sigma(\mathbf{w})(\sigma(\mathbf{w})+\mu(\mathbf{w}))\exp\left(-\frac{\mu(\mathbf{w})}{\sigma(\mathbf{w})}\right).$

####  5.2.2 Relation to limit sets

It is useful to introduce the concept of limit sets and show how these are
related to SPAR models on Laplace margins.

######  Definition 19 (Limit set).

Let $\mathbf{X}_{1},\dots,\mathbf{X}_{n}$ be a sequence of independent and
identically distributed random vectors in $\mathbb{R}^{d}$. Suppose that for
some non-random compact set $G\in\mathbb{R}^{d}$, there is a sequence of real
constants $r_{n}>0$, with $r_{n}\to\infty$ as $n\to\infty$, such that as
$n\to\infty$,

$G_{n}=\\{\mathbf{X}_{1}/r_{n},\cdots,\mathbf{X}_{n}/r_{n}\\}\xrightarrow{P}G,$

with respect to the Hausdorff metric, and $\xrightarrow{P}$ denotes
convergence in probability. Then $G$ is referred to as the limit set of the
scaled sample cloud [19].

[37] showed that when the limit set exists, it is compact and star-shaped. The
limit set can be used to describe various extremal dependence properties of
$\mathbf{X}_{j}$ [38, 39]. Moreover, [20] showed how various dependence
coefficients and representations for multivariate extremes could be related to
properties of the limit set, including those of Ledford and Tawn [11, 12],
[5], and [25]. We will not repeat the details here, but note that SPAR
representations can also be related to these methods, through the link to the
limit set, described below.

Limit sets can be viewed as a multivariate analogue to the scaling of
univariate maxima. Consider a variable $X_{E}$ which follows an exponential
distribution with scale parameter $\sigma$, with cdf
$F_{X_{E}}(x)=1-\exp(-x/\sigma)$, $x\geq 0$. Define $M_{n}$ to be the maximum
of the maximum of $n$ independent variables with cdf $F_{X_{E}}$. As
$n\to\infty$, the distribution of $M_{n}$ converges to a Gumbel distribution
with scale $\sigma$ and location $\sigma\log(n)$, i.e.
$[F_{X_{E}}(x)]^{n}\sim\exp(-\exp(-(x-\sigma\log(n))/\sigma))$, as
$n\to\infty$. Hence the distribution of scaled maxima $M_{n}/\log(n)$
converges to a Gumbel distribution with scale $\sigma/\log(n)$ and location
$\sigma$. Since $\sigma/\log(n)\to 0$ as $n\to\infty$, the limit distribution
is degenerate and we have
$\lim_{n\to\infty}\Pr(|M_{n}/\log(n)-\sigma|>\varepsilon)=0$ for all
$\varepsilon>0$, and hence $M_{n}/\log(n)$ converges in probability to
$\sigma$. In the multivariate setting, the boundary of the limit set can be
viewed as the radius of scaled maxima from the conditional radial
distribution.viiviiviiIn the multivariate context, the observations are
distributed over angle. For a small $\epsilon$-neighbourhood around a
particular angle $\mathbf{w}$, the mean number falling within this
neighbourhood converges to $m=n\epsilon V_{d}f_{\mathbf{W}}(\mathbf{w})$ as
$\epsilon\to 0^{+}$, where $V_{d}$ is the volume of the $d$-dimensional
$L^{1}$ sphere. However, if we set $\epsilon=1/\log(n)$ then
$\log(m)/\log(n)\to 1$ as $n\to\infty$, and we obtain a similar convergence of
scaled maxima at any particular angle. The following proposition, which is an
adaptation of Proposition 2.2 in [20], makes this explicit.

######  Proposition 5.6.

Suppose that the random vector $\mathbf{X}\in\mathbb{R}^{d}$ has standard
Laplace margins and has joint density function $f_{\mathbf{X}}$ that satisfies

$f_{\mathbf{X}}(r\mathbf{x})\sim\mathcal{M}(\exp(-r),\mathbf{x})\exp\left(-\frac{r}{\sigma(\mathbf{x})}\right),\quad
r\to\infty,$ (28)

for $\mathbf{x}\in\mathbb{R}^{d}\setminus\\{\mathbf{0}\\}$, where
$\mathcal{M}(t,\mathbf{w})$ is slowly-varying in $t$ at $0^{+}$ and the
function $\sigma$ is positive, continuous and homogeneous of order $-1$.
Define $G_{n}=\\{\mathbf{X}_{1}/\log(n),\dots,\mathbf{X}_{n}/\log(n)\\}$ to be
a sequence of scaled random samples from $f_{\mathbf{X}}$. Then the following
results hold.

  1. (i)

The scaled sample cloud, $G_{n}$, has a limit set, $G$, with boundary
$\partial
G=\\{\mathbf{w}\,\sigma(\mathbf{w}):\mathbf{w}\in\mathcal{U}_{1}\\}$.

  2. (ii)

The limit set is bounded by the unit box,
$G\subseteq\\{\mathbf{x}\in\mathbb{R}^{d}:\|\mathbf{x}\|_{\infty}=1\\}$, and
this bound is reached in each dimension, i.e. for $j=1,...,d$,

$\displaystyle\max\\{x_{j}:(x_{1},...,x_{d})\in G\\}$ $\displaystyle=1,$
$\displaystyle\min\\{x_{j}:(x_{1},...,x_{d})\in G\\}$ $\displaystyle=-1.$

Clearly, both SPAR model (4) and modified SPAR model (27), which have constant
shape parameter $\xi(\mathbf{w})=0$ and scale function $\sigma(\mathbf{w})$,
satisfy the assumptions of Proposition 5.6. Therefore, the SPAR framework
provides a rigorous means of estimating limit sets when they exist, since the
GP scale parameter is equal to the radius of the limit set. A similar approach
was proposed by [40], where a large quantile of the conditional radial
distribution was used as a proxy for the limit set, estimated by fitting a GP
distribution to the tail of the conditional radial distribution (see also
[41]). However, part (i) of Proposition 5.6 shows explicitly that the $L^{1}$
radius of points on the boundary of limit set is described by the scale
function of the SPAR model, since
$\|\mathbf{w}\sigma(\mathbf{w})\|_{1}=\sigma(\mathbf{w})$ for
$\mathbf{w}\in\mathcal{U}_{1}$. Part (ii) provides bounds on the scale
function in these cases, namely that for
$\mathbf{w}=(w_{1},\dots,w_{d})\in\mathcal{U}_{1}$ we have
$0<|w_{j}|\sigma(\mathbf{w})\leq 1$ for $j=1,\dots,d$ and that the upper bound
is achieved for some $|w_{j}|\leq 1/2$. The upper and lower bounds in part
(ii) are simply a consequence on the choice of margins, for which scaled
maxima and minima must converge to one, as described above.

In Section 4.3 it was mentioned that for two-dimensional copula densities
which have both an ARL and ARE representation, when $\kappa_{(u_{0},v_{0})}=1$
the function $\lambda_{(u_{0},v_{0})}(1-w,w)$ is not differentiable at
$w=1/2$. The reason for this is now evident. Namely, when
$\kappa_{(u_{0},v_{0})}=1$ we have $\lambda_{(u_{0},v_{0})}(1/2,1/2)=1/2$ and
hence the $L^{1}$ radius of the limit set is
$1/\lambda_{(u_{0},v_{0})}(1/2,1/2)=2$ in the direction
$((-1)^{u_{0}+1},(-1)^{v_{0}+1})$. Since the limit set is bounded by the unit
box, and the upper bound is achieved in the corner
$((-1)^{u_{0}+1},(-1)^{v_{0}+1})$, the function
$\lambda_{(u_{0},v_{0})}(1-w,w)$ must have a cusp at $w=1/2$, and hence is not
differentiable at this point.

Again returning to the question of coordinate systems discussed in Theorem
3.3, we can see that when a joint density on Laplace margins has asymptotic
form (28), the coordinate system required to obtain AI angular and radial
variables can be defined in terms of the gauge function of the limit set.

Finally, we note that there are copulas for which the limit set on Laplace
margins is degenerate in certain regions, i.e. the boundary has zero radius at
certain angles. Examples of this type are EV copulas with Hüsler-Reiss
dependence model and the bivariate exponential copula, discussed in Examples
11 and 12 below. We will show that in these cases, the copula still has a SPAR
representation, despite the limit set being degenerate.

####  5.2.3 Bivariate examples

In the bivariate examples below, we will use the notation
$\mathbf{w}=(w_{1},w_{2})=(\cos_{1}(q),\sin_{1}(q))$ and
$\sigma(q)=1/\lambda(\mathbf{w})$ for $q\in(-2,2]$. We will demonstrate that
SPAR models on Laplace margins can represent a range of tail dependence
levels. From the discussion in Section 4.2, when $\delta_{L}(r,\mathbf{w})$
has asymptotic form (25) and also has an ARL representation, the tail orders
in each quadrant are given by
$\kappa_{(u_{0},v_{0})}=\lambda((-1)^{u_{0}+1},(-1)^{v_{0}+1})$. Throughout
this section we use the modified SPAR model (27) to approximate the joint
density. In the examples below, the angular density and thresholds have been
calculated using numerical integration in most cases. Example 3 considered
SPAR models for bivariate Gaussian and t distributions on their own respective
margins. In examples 9 and 10, we revisit these examples and show that
Gaussian and t copulas also have SPAR representations on Laplace margins.
Extreme value copulas have interesting properties so we consider this family
of copulas in some detail. Expressions for the copulas and copula densities
presented here can be found in reference works, e.g. [6].

######  Example 7 (Frank copula).

The Frank copula density is given by

$\displaystyle c(u,v)$ $\displaystyle=-\alpha
h(1)\frac{1+h(u+v)}{(h(u)h(v)+h(1))^{2}},$

for $(u,v)\in[0,1]^{2}$ and $\alpha\in\mathbb{R}\setminus\\{0\\}$, where
$h(z)=e^{-\alpha z}-1$. It is straightforward to show that
$c(u,v)\in(0,\infty)$ for $(u,v)\in[0,1]^{2}$, and hence
$\delta_{L}(r,\mathbf{w})$ is asymptotically a function of $\mathbf{w}$ only,
and therefore of form (25) with $\beta(\mathbf{w})=0$,
$\lambda(\mathbf{w})=1$. This corresponds to asymptotic independence in each
quadrant. The angular density for various values of $\alpha$ is shown in
Figure 11 (top row). A comparison of the joint density from the SPAR model
(dashed black lines) with the true joint density (solid coloured lines) is
shown for a case with $\alpha=10$ and threshold exceedance probability
$\zeta=10^{-4}$. In this case, the modified SPAR model (27) is asymptotically
exact. However, a low threshold exceedance probability is required to give
good agreement between the SPAR density and true density, due to slow
convergence of $\delta_{L}(r,\mathbf{w})$ to the limit in the 2nd and 4th
quadrants of the plane. Any copula with continuous density which is positive
along the edges, has a SPAR representation on Laplace margins, determined
solely by the angular density $f_{Q}(q)$. Other examples include Plackett,
Ali-Mikhail-Haq, and Farlie-Gumbel-Morgenstern copulas. Note that for these
cases the limit sets are all the same, but the SPAR models differ, due to the
different angular densities. $\blacksquare$

######  Example 8 (Joe copula).

The Joe copula density is given by

$\displaystyle c(1-u,1-v)$
$\displaystyle=(uv)^{\alpha-1}z^{\tfrac{1}{\alpha}-2}\left(z+\alpha-1\right),$

where $z=u^{\alpha}+v^{\alpha}-(uv)^{\alpha}$, $(u,v)\in[0,1]^{2}$ and
$\alpha\geq 1$. The Joe copula with $\alpha=1$ is equal to the independence
copula. The dependence function has asymptotic form (25) with
$\beta(\mathbf{w})=0$ and for $\alpha>1$ we have

$\lambda(\mathbf{w})=\begin{cases}1,&q\in(-2,-1]\\\
1+(\alpha-1)|w_{1}|,&q\in(-1,0]\\\
1-\alpha+(2\alpha-1)\max(|w_{1}|,|w_{2}|),&q\in(0,1]\\\
1+(\alpha-1)|w_{2}|&q\in(1,2].\end{cases}$

The angular density, scale parameter and corresponding limit sets are shown in
Figure 11 for various values of $\alpha$. When $\alpha>1$ the distribution is
asymptotically dependent in the 1st quadrant and negatively dependent in the
2nd and 4th quadrants. The distribution is asymptotically quadrant independent
in the 3rd quadrant for all values of $\alpha$. A comparison of the joint
density with the SPAR approximation is shown in Figure 11 for $\alpha=3$ and
threshold exceedance probability $\zeta=0.05$. The agreement is good in all
quadrants. $\blacksquare$

######  Example 9 (Gaussian copula).

The Gaussian copula density, with Pearson correlation $\rho\in(-1,1)$, is
given by

$c(u,v)=\frac{1}{\sqrt{1-\rho^{2}}}\exp\left(-\frac{\rho^{2}(x^{2}+y^{2})-2\rho
xy}{2(1-\rho^{2})}\right),$

for $(u,v)\in[0,1]^{2}$, where $x=\Phi^{-1}(u)$ and $y=\Phi^{-1}(v)$, and
$\Phi^{-1}$ is the inverse of the standard normal cdf.
$\delta_{L}(r,\mathbf{w})$ has asymptotic form (25) with

$\lambda(\mathbf{w})=\frac{1-2\rho\,\mathrm{sgn}(w_{1}w_{2})\sqrt{|w_{1}w_{2}|}}{1-\rho^{2}},$

and $\beta(\mathbf{w})=(\lambda(\mathbf{w})-1)/2$ (see Appendix B.2). The
modified SPAR model for the Gaussian copula on Laplace margins is therefore
not asymptotically exact. However, an approximation using
$\beta(\mathbf{w})=0$ leads to a reasonably good agreement, as shown in Figure
11. In the example shown, $\rho=0.6$ and $\zeta=0.01$. The angular density,
scale parameter and corresponding limit sets are also shown in Figure 11 for
various values of $\rho$. $\blacksquare$

######  Example 10 (t copula).

The t copula density, with Pearson correlation $\rho\in(-1,1)$ and $\nu>0$
degrees of freedom is given by

$c(u,v)=\left(\frac{\Gamma(\nu/2)}{\Gamma((\nu+1)/2)}\right)^{2}\,\frac{\nu}{2\sqrt{1-\rho^{2}}}\,\left(1+\frac{x^{2}+y^{2}-2\rho
xy}{\nu(1-\rho^{2})}\right)^{-\nu/2-1}\,\left(\left(1+\frac{x^{2}}{\nu}\right)\left(1+\frac{y^{2}}{\nu}\right)\right)^{(\nu+1)/2},$

for $(u,v)\in[0,1]^{2}$, where $x=F_{t}^{-1}(u;\nu)$, $y=F_{t}^{-1}(v;\nu)$
and $F_{t}^{-1}(\cdot;\nu)$ is the inverse cdf of the univariate t
distribution on $\nu$ degrees of freedom. $\delta_{L}(r,\mathbf{w})$ has
asymptotic form (25) with $\beta(\mathbf{w})=0$ and (see Appendix B.3)

$\lambda(\mathbf{w})=\max(|w_{1}|,|w_{2}|)+\frac{1}{\nu}||w_{1}|-|w_{2}||.$

In this case, the SPAR model is asymptotically exact. The scale parameter and
corresponding limit sets are independent of the correlation coefficient
$\rho$. However, the angular density is a function of both $\nu$ and $\rho$.
The example shown in Figure 11 for the joint density has $\rho=0.6$, $\nu=2$
and threshold exceedance probability $\zeta=0.05$, the same case as considered
in Example 3 and shown in Figure 5. On Laplace margins, the agreement between
the SPAR model and the true density is slightly better in the first and third
quadrants, than in the 2nd and 4th. $\blacksquare$

Figure 11: SPAR models for Frank (top row), Joe (2nd row), Gaussian (3rd row)
and t (bottom row) copulas on Laplace margins. Left column: angular density
for various values of dependence parameters $\alpha$, $\rho$ and $\nu$. Second
column: Gamma scale parameter. Third column: Limit sets. Fourth column:
Isodensity contours for joint density (coloured lines) and SPAR approximation
(black dashed lines). Contours are at density levels $10^{-2}$, $10^{-4}$,
$10^{-6}$, etc.

######  Example 11 (Extreme value copulas).

As mentioned above, EV copulas have ARE form in the lower tail and hence from
Propositions 4.3 and 5.5, they satisfy the SPAR model assumptions for the
negative orthant of $\mathbb{R}^{d}$. In other orthants, it is useful to
consider some specific examples to illustrate various possibilities. Appendix
B.1 considers two-dimensional cases where the stable tail dependence function
follows either the symmetric logistic, asymmetric logistic, or Hüsler-Reiss
form. Here, we summarise results that are of general interest and refer to
Appendix B.1 for details.

For the case of symmetric logistic dependence with parameter $\alpha>1$, we
find that $\delta_{L}(r,\mathbf{w})$ has asymptotic form (25), with
$\beta(\mathbf{w})=0$ for $q\in(-2,-1)\cup[0,1]$ and
$\beta(\mathbf{w})=1-\alpha$ for $q\in[-1,0)\cup(1,2]$. The angular density,
GP scale parameter and limit sets are shown in Figure 12 for various values of
$\alpha$. A comparison of the SPAR approximation with the true density is also
shown, for the case $\alpha=2$ and $\zeta=0.05$. The agreement is good in the
first and third quadrants, but slightly worse in the second and fourth due to
the non-zero value of $\beta$ for this angular range.

For the case of the asymmetric logistic model [42], we find that
$\delta_{L}(r,\mathbf{w})$ has asymptotic form (25), with
$\beta(\mathbf{w})=0$ for all $q$. The limit set is symmetric about the line
$q=1/2$. However, at finite levels, the corresponding isodensity contours are
not symmetric about $q=1/2$, as illustrated in Figure 12, for a case with
$\alpha=5$, $\gamma_{1}=0.9$ and $\gamma_{2}=0.1$. To improve the agreement
between the SPAR approximation and true joint density at finite levels, we can
define the angular-radial coordinate system relative to a different origin. A
finite shift in the origin does not affect the asymptotic behaviour of
$\delta_{L}(r,\mathbf{w})$, and hence the GP scale and shape parameters for
the SPAR approximation, but can improve the rate of convergence of the true
density to the SPAR approximation. The SPAR approximation for a case with
parameters $\alpha=5$, $\gamma_{1}=0.9$ and $\gamma_{2}=0.1$ is shown in
Figure 12, with the origin placed at $(x_{0},y_{0})=(-0.859,-3.3)$. The
rationale for this choice of origin is discussed in Appendix B.1, where it is
shown to improve the speed of convergence of $\delta_{L}(r,\mathbf{w})$ to its
asymptotic form. In the first quadrant, the threshold has been set as
$\mu(q)=\max\\{10,|x_{0}/\cos_{1}(q)|,|y_{0}/\sin_{1}(q)|\\}$, to ensure good
agreement between the asymptotic SPAR approximation and the true density. The
values $|x_{0}/\cos_{1}(q)|$ and $|y_{0}/\sin_{1}(q)|$ correspond to the
positive x- and y-axes of the original Cartesian coordinate system, and the
minimum value of 10 has been chosen somewhat arbitrarily through trial and
error so that the asymptotic approximation is reasonable. In the third
quadrant, the density is strictly decreasing for all $q\in[-2,-1]$ and the
threshold is set at a constant value $\mu(q)=10$. Similarly, we set
$\mu(q)=\max\\{10,|y_{0}/\sin_{1}(q)|\\}$ in the second quadrant, and
$\mu(q)=\max\\{10,|x_{0}/\cos_{1}(q)|\\}$ in the fourth quadrant. The
corresponding threshold exceedance probabilities are then calculated using
numerical integration. In the regions where the SPAR model is defined, the
agreement with the true density contours is good, including the region along
the higher-density ‘finger’ in the first quadrant.

Finally, for the case of the Hüsler-Reiss dependence model [35], with
parameter $\alpha>0$, we find that $\delta_{L}(r,\mathbf{w})$ does not have
asymptotic form (25) or a continuous limit set. Instead, we find that
$\delta_{L}(r,\mathbf{w})\propto\exp\left(-\tfrac{1}{2}\left[\left(\tfrac{\alpha}{2}(w_{1}-w_{2})r\right)^{2}-r\right]\right)$
for $q\in(0,1)$. However, it is shown in Appendix B.1 that the SPAR
assumptions (A1)-(A3) are nevertheless satisfied for this copula. This
illustrates that the conditions of Proposition 5.5 are sufficient but not
necessary for a copula to have a SPAR representation on Laplace margins.
Moreover, this example shows that the SPAR approach provides a more general
framework for representing multivariate extremes than using limit sets alone.

In summary, we find that the lower tail of EV copulas have a ‘natural’
representation on Laplace margins, but that the case for other orthants is
more complicated. In Section 5.3 it will be shown that the upper tails of EV
copulas have a natural representation on heavy-tailed margins, but that there
are severe limitations on the types of copula that have SPAR representations
on these margins. $\blacksquare$

Figure 12: As previous figure, but for extreme value copulas with symmetric
logistic dependence (top row) and asymmetric logistic dependence (bottom row)
on Laplace margins. For EV asymmetric logistic copula, red dot indicates
origin of angular-radial coordinates, red line indicates SPAR threshold.

######  Example 12 (Bivariate exponential copula).

This example illustrates that although the tail order is infinite in the lower
left corner, and the limit set is degenerate in this quadrant, the bivariate
exponential copula does have a SPAR representation on Laplace margins. The
bivariate exponential copula with parameter $\alpha\in[0,1]$ is given by

$C(u,v)=uv\exp(-\alpha\log(u)\log(v)),$

for $(u,v)\in[0,1]^{2}$, with corresponding density function

$c(u,v)=\exp(-\alpha\log(u)\log(v))\left(\alpha^{2}\log(u)\log(v)-\alpha\log(uv)-\alpha+1\right).$

The case $\alpha=0$ corresponds to the independence copula. For
$\alpha\in(0,1]$ and $x>0$, in the lower left corner we have

$\displaystyle\lim_{t\to 0^{+}}\frac{C(x(t,t))}{C(t,t)}=\lim_{t\to
0^{+}}\frac{(xt)^{2-\alpha\log(xt)}}{t^{2-\alpha\log(t)}}=\begin{cases}0,&x<1,\\\
1,&x=1,\\\ \infty,&x>1.\end{cases}$

Hence the lower tail order is $\kappa_{(0,0)}=\infty$. For $\alpha\in(0,1)$,
the angular-radial representation of the copula density has asymptotic form

$\displaystyle\delta_{L}(r,\mathbf{w})$
$\displaystyle\sim\begin{cases}2^{-\alpha\log(2)}\alpha r\left[\alpha
w_{1}w_{2}r+\alpha\log(2)+1\right]\exp(-\alpha[r^{2}w_{1}w_{2}+r\log(2)]),&q\in[-2,-1],\\\
\alpha|w_{2}|r,&q\in(-1,0),\\\ 1-\alpha+\alpha\log(2),&q=0,1,\\\
1-\alpha,&q\in(0,1),\\\ \alpha|w_{1}|r,&q\in(1,2).\end{cases}$

This does not have asymptotic form (25) in the lower left quadrant. However,
assumptions (A1) and (A2) are satisfied, taking $\xi(q)=0$ and
$\sigma(\mu,q)=1$ for $q\in[-1,2]$ and $\sigma(\mu,q)=(1+2\alpha
w_{1}w_{2}\mu)^{-1}$ for $q\in(-1,-2)$. When $\alpha=1$, the asymptotic form
of $\delta_{L}(r,\mathbf{w})$ is the same for $q\in(-2,0]\cup[1,2]$, whereas
for $q\in(0,1)$ we have $\delta_{L}(r,\mathbf{w})\sim
g(\mathbf{w})\exp(-r\min(w_{1},w_{2}))$, where $g(\mathbf{w})=1$ for
$q=\tfrac{1}{2}$ and $g(\mathbf{w})=\tfrac{1}{2}$ otherwise. For any value of
$\alpha\in[0,1]$, the angular density is continuous and finite and hence
satisfies assumption (A3). $\blacksquare$

###  5.3 SPAR models on heavy-tailed margins

When considering joint extremes on heavy-tailed margins, it is common to use
the framework of multivariate regular variation (MRV) [28] and hidden regular
variation (HRV) [43]. However, [30] showed that an MRV distribution can be
constructed from a copula with upper tail order $\kappa_{\mathbf{1}_{d}}=1$,
coupled with heavy-tailed margins (i.e. margins in the domain of attraction of
an extreme value distribution with positive shape parameter). Similarly, [31]
showed that an HRV distribution can be constructed from a copula with upper
tail order $\kappa_{\mathbf{1}_{d}}\in(1,d)$, coupled with heavy-tailed
margins. Therefore, for consistency with previous sections, we continue to use
the framework of tail order functions and tail density functions and the ARL
model.

Proposition 4.4 showed that the AR copula function on GP margins can be
derived in terms of the ARL representation of the copula density in the upper
corner, when this exist. Since the relation in Proposition 4.4 is valid for
any $\xi_{m}>0$, for simplicity we will assume $\xi_{m}=1$ in the following.
Moreover, as discussed in Section 5.1, the use of Pareto margins with the
polar origin defined at $\mathbf{x}=\mathbf{1}_{d}$ is equivalent to using GP
margins with the polar origin defined at $\mathbf{x}=\mathbf{0}_{d}$, but
Pareto margins are more convenient for visualisations on a log-log scale.
Therefore, throughout this section, we will work with Pareto margins with the
polar origin defined at $\mathbf{x}=\mathbf{1}_{d}$.

######  Proposition 5.7.

Suppose that $\mathbf{X}\in[1,\infty)^{d}$ is a random vector with standard
Pareto margins and copula density $c$. Suppose also that $c$ satisfies the ARL
model assumptions in the upper and lower tails and is continuous and finite
away from the corners. Define $R=\|\mathbf{X}-\mathbf{1}_{d}\|_{1}$ and
$\mathbf{W}=(\mathbf{X}-\mathbf{1}_{d})/R$. Then

$f_{R,\mathbf{W}}(r,\mathbf{w})\sim\tilde{b}(\mathbf{w})\mathcal{L}_{\mathbf{1}_{d}}\left(r^{-1}\right)r^{-1-\kappa_{\mathbf{1}_{d}}},\quad
r\to\infty,$

where
$\tilde{b}(\mathbf{w})=\left[\prod_{j=1}^{d}w_{j}^{-2}\right]s_{\mathbf{w}}^{\kappa_{\mathbf{1}_{d}}-d}b_{\mathbf{1}_{d}}\left((s_{\mathbf{w}}\mathbf{w})^{-1}\right)$
and $s_{\mathbf{w}}=\sum_{j=1}^{d}w_{j}^{-1}$. Moreover, the joint density
$f_{R,\mathbf{W}}(r,\mathbf{w})$ satisfies assumptions (A1)-(A3) for
$\mathbf{w}\in\mathcal{U}_{1}\cap(0,1]^{d}$, with
$\xi(\mathbf{w})=1/\kappa_{\mathbf{1}_{d}}$ and
$\sigma(\mu,\mathbf{w})=\mu/\kappa_{\mathbf{1}_{d}}$.

The expression for $f_{R,\mathbf{W}}$ on Pareto margins, given in Proposition
5.7 is equivalent to that obtained from the Ledford-Tawn model [11, 12] in
Cartesian coordinates. This is a direct consequence of the formulation of the
ARL model for the copula, discussed by [7]. Proposition 5.7 therefore gives a
relation between the Ledford-Tawn model and SPAR models on Pareto margins.
However, Proposition 5.7 only tells us that the SPAR assumptions are satisfied
in the joint exceedance region, where all variables are large, since we have
assumed that $\mathbf{w}=(w_{1},...,w_{d})\in\mathcal{U}_{1}\cap(0,1]^{d}$. In
the remainder of the section, we will examine the circumstances in which the
SPAR assumptions are satisfied for $w_{j}\to 0^{+}$ for one or more
$j\in\\{1,...,d\\}$.

Before we consider this further, there is another important point to note
regarding the asymptotic GP scale parameters. Proposition 5.7 gives a scaling
function $\sigma(\mu,\mathbf{w})$ for the asymptotic GP distribution that is
independent of $\mathbf{w}$. However, it is important to note that whilst the
conditional radial tail distribution is asymptotically equivalent to
$\bar{F}_{GP}(r;1/\kappa_{\mathbf{1}_{d}},\mu/\kappa_{\mathbf{1}_{d}})$, this
is not necessarily the best approximation at finite levels. Consider the case
of a GP distribution, with scale $\sigma_{0}$ and shape $\xi>0$. In this case
we have

$\frac{\bar{F}_{GP}(\mu+r;\xi,\sigma_{0})}{\bar{F}_{GP}(\mu;\xi,\sigma_{0})}=\frac{\left(1+\xi\dfrac{\mu+r}{\sigma_{0}}\right)^{-1/\xi}}{\left(1+\xi\dfrac{\mu}{\sigma_{0}}\right)^{-1/\xi}}=\left(1+\xi\frac{r}{\sigma_{0}+\xi\mu}\right)^{-1/\xi}=\bar{F}_{GP}(r;\xi,\sigma_{0}+\xi\mu).$

That is, the tail distribution of a GP distribution at threshold level $\mu$
is also a GP distribution, with the same shape parameter and scale parameter
$\sigma_{0}+\xi\mu$. This is the well-known threshold stability property of
the GP distribution. However,
$\bar{F}_{GP}(r;\xi,\sigma_{0}+\xi\mu)\sim\bar{F}_{GP}(r;\xi,\xi\mu)$ as
$\mu\to\infty$. So, the Pickands-Balkema-de Haan theorem would be satisfied
taking $\sigma(\mu)=\mu\xi$, whereas a more accurate approximation to the tail
(an equality in this case) is obtained by taking
$\sigma(\mu)=\sigma_{0}+\xi\mu$. When $\sigma_{0}$ is large relative to
$\xi\mu$, neglecting this term can lead to large errors in the GP
approximation at finite levels. The next example illustrates that in some
bivariate cases $\sigma_{0}\to\infty$ as $\mathbf{w}\to(0,1)$.

######  Example 13 (EV copulas on Pareto margins).

For EV copulas we have $c_{\mathbf{1}_{d}}(t\mathbf{w})\sim
t^{1-d}\left|A^{(1,...,1)}(\mathbf{w})\right|$, where $A$ is the stable tail
dependence function (see Appendix B.1). In the two-dimensional case, for
random vector $(X,Y)\in[1,\infty)^{2}$, we define $R=(X-1)+(Y-1)$ and
$Q=(Y-1)/R$. Recalling that in two-dimensions
$f_{R,\mathbf{W}}(r,(1-q,q))=f_{R,Q}(r,q)$ (see discussion in Section 3.2),
from Proposition 5.7 we have

$f_{R,Q}(r,q)\sim\tilde{b}(q)r^{-2},\quad r\to\infty,$ (29)

where $\tilde{b}(q)=-(q(1-q))^{-1}A^{(1,1)}(q,1-q)$. Note that
$f_{GP}(r;1,\sigma)\sim\sigma r^{-2}$ for $r\to\infty$. Hence, for the
conditional radial density we have

$\displaystyle f_{R|Q}(r|q)$ $\displaystyle\sim
f_{GP}\left(r;1,\sigma_{0}(q)\right),\quad r\to\infty,$

where $\sigma_{0}(q)=\tilde{b}(q)/f_{Q}(q)$. Hence, from the discussion above
a more accurate approximation to the tail of the radial distribution at
threshold $\mu$, than given in Proposition 5.7, is a GP distribution with
shape $\xi(q)=1$ and scale $\sigma(\mu,q)=\mu+\sigma_{0}(q)$. To investigate
the behaviour of $\sigma_{0}(q)$ for $q\to 0^{+}$, we revisit the three models
for the stable tail dependence function which were considered in Example 11
(see Appendix B.1 for details).

For the case of symmetric logistic dependence with parameter $\alpha\geq 1$,
we find that $f_{Q}(q)\to 0^{+}$ as $q\to 0^{+}$ for all $\alpha>1$. When
$\alpha=1$, the copula is equal to the independence copula and hence
$f_{Q}(q)\to\infty$ as $q\to 0^{+}$. When $\alpha>1$,
$\tilde{b}(q)=(\alpha-1)(q(1-q))^{\alpha-2}(q^{\alpha}+(1-q)^{\alpha})^{\tfrac{1}{\alpha}-2}$
and hence

$\tilde{b}(q)\to\begin{cases}\infty,&\alpha<2,\\\ 1,&\alpha=2,\\\
0,&\alpha>2,\end{cases}\qquad q\to 0^{+}.$

Therefore $\sigma_{0}(q)\to\infty$ as $q\to 0^{+}$ when $\alpha\leq 2$. So, in
this case the assumption of a finite scale parameter function (A2) does not
hold for $q\to 0^{+}$. When $\alpha>2$, numerical calculations indicate that
$\sigma_{0}(q)\to 0^{+}$ as $q\to 0^{+}$, so for these cases the copula
appears to have a valid SPAR representation for all $q\in[0,1]$. Examples of
the angular density and GP scale parameter as a function of $q$ are shown in
Figure 13 for various values of $\alpha$. Contour plots of the joint densities
$f_{R,Q}$ and $f_{X,Y}$ are also shown for the case $\alpha=3$ together with
the SPAR approximations. For the Cartesian density, the agreement is good both
in the joint exceedance region and close to the margins.

For the case of asymmetric logistic dependence, the angular density goes to
infinity as $q\to 0^{+}$ or $q\to 1^{-}$ for any $\alpha\geq 1$ and
$\gamma_{1},\gamma_{2}\in(0,1)$. Hence the SPAR model assumptions are only
valid in the joint exceedance region $0<q<1$. Examples of the joint densities
$f_{R,Q}$ and $f_{X,Y}$ are shown in Figure 13 for the case $\alpha=5$,
$\gamma_{1}=0.1$, $\gamma_{2}=0.9$ (the same case considered on Laplace
margins), together with the SPAR approximations. For the Cartesian density,
the location of the ‘finger’ is captured by the SPAR approximation, without
the need for a judicious choice of origin, required on Laplace margins. The
reason for this is that the asymmetric behaviour of the joint density is
contained in the partial derivative of the stable tail dependence function
$A^{(1,1)}$, and hence this information is also represented in
$\sigma_{0}(q)$. In contrast, $\delta_{L}(r,\mathbf{w})$ is symmetric, so the
asymmetric behaviour at finite levels must be accounted for using a shift of
origin on Laplace margins. However, on Pareto margins, the density close to
the axes is not captured by the SPAR approximation. In the angular-radial
space, the SPAR approximation appears much better, since for any fixed
$q\in(0,1)$ the SPAR approximation converges to the true density as
$r\to\infty$.

Finally, for the Hüsler-Reiss dependence model, we find that $f_{Q}(q)\to
0^{+}$ and $\sigma_{0}(q)\to 0^{+}$ for $q\to 0^{+}$ or $1^{-}$ for any
$\alpha>0$ – see Figure 13. So the SPAR assumptions are valid close to the
margins as well. An example of the SPAR approximation to the joint density for
the case $\alpha=1$ is shown in Figure 13. The SPAR approximation is in good
agreement with the true density both in the joint exceedance region and close
to the margins.

Figure 13: SPAR models for EV copulas on standard Pareto margins, with polar
origin at $(x,y)=(1,1)$, for symmetric logistic dependence (top row),
asymmetric logistic dependence (middle row) and Hüsler-Reiss dependence model
(bottom row). First and second columns show angular density and GP scale
parameter for various dependence parameter values, $\alpha$. Third column
shows logarithmically-spaced contours of the true angular-radial joint density
(coloured lines) and SPAR approximation (black dashed lines). Right hand
column shows logarithmically-spaced contours of the true Cartesian joint
density (coloured lines) and SPAR approximation (black dashed lines).

These examples illustrate that while SPAR models provide a good approximation
to the upper tail of EV copulas on Pareto margins in the joint exceedance
region, they do not necessarily provide a good approximation close to the
axes, since either the angular density or scale parameter can become unbounded
for $q\to 0^{+}$ or $1^{-}$. The angular-radial representation of the joint
density in (29), is the same as that derived by [33]. So from a probabilistic
standpoint, the SPAR model for EV copulas on Pareto margins is equivalent to
the Coles-Tawn method. However, from a statistical perspective the SPAR
approach is less parsimonious in this application, since it requires
estimation of both the angular density and GP scale parameter, whereas
$\tilde{b}(q)$ is estimated directly by fitting parametric models in the
Coles-Tawn method. $\blacksquare$

Proposition 4.1 showed that in certain cases with $\kappa_{\mathbf{1}_{d}}>1$,
the ARL representation for the copula was only valid in a limited region. The
next proposition makes use of this result together with Proposition 5.7, to
show that for these cases, SPAR models on Pareto margins are only valid in the
joint exceedance region.

######  Proposition 5.8.

Suppose that random vector $(X,Y)\in[1,\infty)^{2}$ has standard Pareto
margins and copula density $c$, and define $R=(X-1)+(Y-1)$ and $Q=(Y-1)/R$.
Suppose also that $c$ satisfies the assumptions of Proposition 4.1 in the
upper right corner, with $\beta=0$ and $\kappa_{(1,1)}>1$. Then for
$q\in(0,1)$ the angular-radial joint density is asymptotically

$f_{R,Q}(r,q)\sim(q(1-q))^{-1-\tfrac{\kappa_{(1,1)}}{2}}\mathcal{L}(r)r^{-1-\kappa_{(1,1)}},\qquad
r\to\infty.$

for some $\mathcal{L}\in RV_{0}(\infty)$.

Under the assumptions of Proposition 5.8, the angular variation of the density
is dependent only on the tail order. For fixed $r$, the approximation for
$f_{R,Q}(r,q)$ given in Proposition 5.8 tends to infinity when $q\to 0^{+}$ or
$1^{-}$, and hence the SPAR representation is only valid in the joint
exceedance region. Examples where the assumptions of Proposition 5.8 are
satisfied include the case where $c(u,v)=c_{EV}(1-u,1-v)$ where $c_{EV}$ is an
extreme value copula density, with symmetric stable tail dependence function,
such as the symmetric logistic model or Hüsler-Reiss model (see Appendix B.1),
and when $c$ is the Gaussian copula (see Appendix B.2). Figure 14 illustrates
these examples, and shows contours of the true joint density together with the
SPAR approximations, both in the angular-radial space and Cartesian space. It
is evident that the asymptotic expression for the angular-radial joint density
given in Proposition 5.8 is valid for any fixed $q\in(0,1)$ as $r\to\infty$.
However, in the Cartesian space with a log-log scale used for the margins, the
approximation is only valid in the region where both variables are large, and
the approximations tend to infinity on the margins.

Figure 14: As previous figure, but for lower tail of EV copula with symmetric
logistic dependence with $\alpha=3$ (top) and Gaussian copula with $\rho=0.6$
(bottom).

In summary, copulas with strong tail dependence have a natural SPAR
representation on heavy-tailed margins in the joint exceedance region.
However, this representation is not always valid close to the margins.
Moreover, for many copulas without strong tail dependence, the SPAR
representation on heavy-tailed margins is only valid in the joint exceedance
region. In general, heavy-tailed margins are less useful than Laplace margins
for SPAR representations.

Finally, although this may be obvious to some readers, it is worth mentioning
that joint densities on heavy-tailed margins do not have limit sets. Consider
the case of GP margins with shape $\xi_{m}>0$ and scale $\sigma=1$. The
distribution of $M_{n}$, the maximum of $n$ samples from the marginal
distribution, converges to a generalised extreme value (GEV) distribution with
shape $\xi$, scale $n^{\xi}$ and location $(n^{\xi}-1)/\xi$, as $n\to\infty$.
Therefore, the distribution of scaled maxima $M_{n}/n^{\xi}$, converges to a
GEV distribution with shape $\xi$, unit scale and location $1/\xi$. This limit
is non-degenerate, and therefore the scaled sample has no upper bound. Since
the scaled margins do not converge to a finite interval, scaled samples from
the joint distribution do not converge onto a finite set.

###  5.4 SPAR models on short-tailed margins

To illustrate some features of SPAR models on short-tailed margins (i.e.
margins in the domain of attraction of an extreme value distribution with
negative shape parameter), it is useful to start by considering the case of
the independence copula. Throughout this section, we will consider two-
dimensional examples and assume that
$(X,Y)\in\mathcal{D}\subset\mathbb{R^{2}}$ has GP margins with unit scale
parameter and shape parameter $\xi_{m}<0$. Angular and radial variables are
defined as $R=X+Y$ and $Q=Y/R$. The upper end point of $X$ and $Y$ is
$-1/\xi_{m}$. The upper end point of $R$ is dependent on $Q$ and the copula of
$(X,Y)$, and will be denoted $r_{F}(q)$ at $Q=q$.

######  Example 14 (Independence on GP margins with negative shape).

Suppose that $(X,Y)$ and $(R,Q)$ are random vectors defined as above, and that
$X$ and $Y$ are independent. The upper end point of $R$ at $Q=q$ is given by
$r_{F}(q)=-(\xi_{m}\max(q,1-q))^{-1}$. The angular-radial joint density is
given by $f_{R,Q}(r,q)=r\,m_{GP}(r,(1-q,q))$, where $m_{GP}$ is the marginal
product function, given below. If we substitute $r=r_{F}(q)-s$, then for
$r\in[0,r_{F}(q)]$ and $q\in[0,1]$ we have

$\displaystyle m_{GP}(r,(1-q,q))$
$\displaystyle=f_{GP}(r(1-q);\xi_{m},1)f_{GP}(rq;\xi_{m},1)$
$\displaystyle=\left[(1+\xi_{m}r(1-q))(1+\xi_{m}rq)\right]^{-1-\tfrac{1}{\xi_{m}}}$
$\displaystyle=\begin{cases}\left[\left(1-\dfrac{\min(q,1-q)}{\max(q,1-q)}-\dfrac{s}{r_{F}(q)}\right)\left(\dfrac{s}{r_{F}(q)}\right)\right]^{-1-\tfrac{1}{\xi_{m}}},&q\in[0,1/2)\cup(1/2,1]\\\
\left(\dfrac{s}{r_{F}(q)}\right)^{-2-\tfrac{2}{\xi_{m}}},&q=1/2.\end{cases}$

The angular density, $f_{Q}(q)$, is finite for $q\in[0,1]$ for all
$\xi_{m}<0$. Moreover, assumption (A1) is satisfied, with $\xi(q)=\xi_{m}$ for
$q\in[0,1/2)\cup(1/2,1]$ and $\xi(1/2)=(\xi_{m}+2)/\xi_{m}$. Hence, unless
$\xi_{m}=-1$, there is a discontinuity in the radial shape parameter and hence
(A2) is not satisfied. In the case that $\xi_{m}=-1$, we have
$m_{GP}(r,(1-q,q))=1$ and $f_{R,Q}(r,q)=r$. So although assumptions (A1)-(A3)
are satisfied, the approximation of the tail of $f_{R|Q}(r|q)$ with a GP
distribution with $\xi=-1$ (i.e. a uniform distribution) is not a good
approximation, since $f_{R|Q}(r|q)$ is linearly increasing in $r$.
$\blacksquare$

Example 14 shows that there is a discontinuity in the domain of attraction of
the marginal product function at $q=1/2$. This arises because for $q<1/2$ the
variable $Y$ does not reach its upper end point as $r\to r_{F}(q)$ and hence
the density $f_{GP}(rq;\xi_{m},1)$ converges to a non-zero value as $r\to
r_{F}(q)$. Similarly, for $q>1/2$ the variable $X$ does not reach its upper
end point as $r\to r_{F}(q)$ (see Figure 10). So for assumption (A2) to be
satisfied on GP margins with negative shape, either a copula must compensate
for this discontinuity, or the support of the copula must be such that
$\delta_{GP}(r,(1-q,q))=0$ for $r<-(\xi_{m}\max(q,1-q))^{-1}$, so that
$m_{GP}(r,(1-q,q))$ is non-zero everywhere within the support and of
$f_{R,Q}$, and the domain of attraction of $f_{R|Q}(r|q)$ is controlled by the
behaviour of $\delta_{GP}(r,(1-q,q))$ at the edge of its support.

For the copulas considered so far in this work, $c(u,v)>0$ for
$(u,v)\in(0,1)^{2}$, so these copulas result in tail behaviour of $f_{R|Q}$
that is governed by both $m_{GP}$ and $\delta_{GP}$. Archimedean copulas with
non-strict generators (see e.g. [44]) provide an example where the Hausdorff
distance between the support of $c$ and $[0,1]^{2}$ can be greater than zero.
Two examples where the SPAR assumptions are satisfied on short-tailed margins
are given below. In both examples, we will establish the parameters of
asymptotic GP distribution by reparameterising the density as follows. The GP
density $f_{GP}(r;\xi,\sigma)$ with $\xi<0$, has upper end point
$r_{F}=-\sigma/\xi$. If we substitute $r=r_{F}-s$, then

$f_{GP}(r_{F}-s;\xi,\sigma)=\sigma^{\tfrac{1}{\xi}}(-\xi
s)^{-1-\tfrac{1}{\xi}},\quad s\in[0,r_{F}].$ (30)

Now suppose that the conditional radial density has asymptotic form

$f_{R|Q}(r_{F}(q)-s|q)\sim g(q)s^{\beta},\quad s\to 0^{+},$

for some continuous function $g(q)>0$ and $\beta>-1$, where $r_{F}(q)$ is the
upper end point of $R$ conditional on $Q=q$. Equating this with the GP
parameters shows that $f_{R|Q}(r|q)\sim f_{GP}(r;\xi,\sigma(q))$ as $r\to
r_{F}(q)=-\sigma(q)/\xi$, with $\xi=-1/(1+\beta)$ and
$\sigma(q)=(g(q))^{\xi}(-\xi)^{1+\xi}$.

######  Example 15 (Clayton copula).

The Clayton copula with parameter $\alpha\in[-1,\infty)\setminus\\{0\\}$ is
given by $C(u,v)=z^{-1/\alpha}$, where $z=\max(u^{-\alpha}+v^{-\alpha}-1,0)$
and $(u,v)\in[0,1]^{2}$. When $\alpha<0$, we have $C(t,t)=0$ for
$t\in[0,2^{1/\alpha}]$. When $\alpha>0$, we have $C(t,t)>0$ for $t>0$. An
example contour plot of the copula is shown in Figure 15 for the case
$\alpha=-0.2$. In the remainder of this example, we assume that
$\alpha\in[-1,0)$. The corresponding density is

$c(u,v)=\begin{cases}(1-\alpha)(uv)^{-\alpha-1}z^{-\tfrac{1}{\alpha}-2},&(u,v)\in\\{(a,b)\in[0,1]^{2}:a^{-\alpha}+b^{-\alpha}\geq
1\\},\\\ 0,&\text{otherwise.}\end{cases}$

Since the support of $c$ is limited in the lower left corner, we work with the
density of the survival copula instead, $c_{(1,1)}(u,v)=c(1-u,1-v)$. On GP
margins with $\xi_{m}<0$, the upper end point of
$\delta_{GP}(r,(1-q,q);c_{(1,1)})$ occurs when $z=0$, or equivalently
$(1+\xi_{m}r(1-q))^{\alpha/\xi_{m}}+(1+\xi_{m}rq)^{\alpha/\xi_{m}}=1$. Setting
$\xi_{m}=\alpha$ leads to a simple solution, with the upper end point of
$\delta_{GP}$ given by $r_{F}(q)=-1/\alpha$ for $q\in[0,1]$. Substituting
$r=r_{F}(q)-s$, we find that the angular-radial joint density is independent
of $q$ and given by

$\displaystyle f_{R,Q}(r,q)$ $\displaystyle=(1+\alpha)r(1+\alpha
r)^{-\tfrac{1}{\alpha}-2}$
$\displaystyle=-(1+\alpha)\left(\frac{1}{\alpha}+s\right)(-\alpha
s)^{-\tfrac{1}{\alpha}-2}.$
$\displaystyle\sim-\left(\frac{1}{\alpha}+1\right)(-\alpha
s)^{-\tfrac{1}{\alpha}-2},\quad s\to 0^{+}$
$\displaystyle=f_{GP}(r_{F}-s;\xi,\sigma),$

where $\xi=\alpha/(1+\alpha)$ and $\sigma=(-\xi)^{-\xi}(1-\xi)^{\xi+1}$. Since
$f_{R,Q}(r,q)$ is independent of $q$, $f_{Q}(q)=1$ and
$f_{R,Q}(r,q)=f_{R|Q}(r|q)$. Hence SPAR model assumptions (A1)-(A3) are
satisfied. The Cartesian joint density $f_{X,Y}(x,y)$ for the case
$\alpha=1/5$ is shown in Figure 15. In this case, the corresponding SPAR
approximation is not shown, as the contours are closely spaced. However, the
discussion above shows that it is asymptotically exact. $\blacksquare$

######  Example 16 (Nelsen copula 4.2.15).

Table 4.2 in [44] presents a list of one-parameter families of Archimedean
copulas. The copula labelled 4.2.15, with parameter $\alpha\geq 1$ is given by
$C(u,v)=z^{\alpha}$ for $(u,v)\in[0,1]^{2}$, where

$z=\max\left(1-\|(x,y)\|_{\alpha},0\right),$

and $x=1-u^{1/\alpha}$, $y=1-u^{1/\alpha}$, and $\|\cdot\|_{\alpha}$ is the
$L^{\alpha}$ norm. We have $C(t,t)=0$ for $t\leq(1-2^{-1/\alpha})^{\alpha}$.
An example contour plot of the copula is shown in Figure 15 for the case
$\alpha=3$. The corresponding density is

$c(u,v)=\begin{cases}\left(1-\tfrac{1}{\alpha}\right)(uv)^{\tfrac{1}{\alpha}-1}(xy)^{\alpha-1}\left(x^{\alpha}+y^{\alpha}\right)^{\tfrac{1}{\alpha}-2}z^{\alpha-2},&(u,v)\in\\{(a,b)\in[0,1]^{2}:\|(1-a^{1/\alpha},1-b^{1/\alpha})\|_{\alpha}\leq
1\\}\\\ 0,&\text{otherwise}.\end{cases}$

As in the previous example, we will work with the density of the survival
copula, $c_{(1,1)}(u,v)=c(1-u,1-v)$. Using GP margins with
$\xi_{m}=-1/\alpha$, the upper end point of $\delta_{GP}(r,(1-q,q);c_{(1,1)})$
occurs when $z=0$. Solving for $r$ shows that the upper end point is given by
$r_{F}(q)=\alpha/\|(1-q,q)\|_{\alpha}$. Making the substitution
$r=r_{F}(q)-s$, we find that the angular-radial joint density is given by

$f_{R,Q}(r_{F}(q)-s,q)=\alpha^{2-\alpha}(\alpha-1)(q(1-q))^{\alpha-1}((1-q)^{\alpha}+q^{\alpha})^{-\tfrac{1}{\alpha}-1}s^{\alpha-2}.$

Integrating this gives the angular density and conditional radial density as

$\displaystyle f_{Q}(q)$
$\displaystyle=\alpha(q(1-q))^{\alpha-1}((1-q)^{\alpha}+q^{\alpha})^{-2},$
$\displaystyle f_{R|Q}(r_{F}(q)-s|q)$
$\displaystyle=\alpha^{1-\alpha}(\alpha-1)((1-q)^{\alpha}+q^{\alpha})^{1-\tfrac{1}{\alpha}}s^{\alpha-2}.$

The conditional radial density $f_{R|Q}$ is exactly of the form (30) and hence
is a GP density, with parameters $\xi=-1/(\alpha-1)$ and
$\sigma(q)=(g(q))^{\xi}(-\xi)^{1+\xi}$, with
$g(q)=\alpha^{1-\alpha}(\alpha-1)((1-q)^{\alpha}+q^{\alpha})^{1-\tfrac{1}{\alpha}}$.
The Cartesian joint density $f_{X,Y}(x,y)$ for the case $\alpha=3$ is shown in
Figure 15, together with the angular density for various values of $\alpha$.
Since the SPAR representation is exact everywhere in the domain, this is not
shown. $\blacksquare$

Figure 15: Examples of Archimedean copulas on GP margins with negative shape
parameter, for Clayton copula with $\alpha=-0.2$ (top row) and Nelsen copula
4.2.15 with $\alpha=3$ (bottom row). Left column: contours of the copula
levels from 0 to 0.9 at increments of 0.1. Thick black line indicates level
set $C(u,v)=0$. Middle column: contours of joint density $f_{X,Y}$ for
corresponding survival copulas on GP margins, with contour levels at
logarithmic increments. Thick black line indicates level set $f_{X,Y}(x,y)=0$.
Right column: angular density for various copula parameter $\alpha$.

The examples presented above are for cases where the angular-radial density
assumes a simple form for a judicious choice of shape parameter for the
margins. Some Archimedean copulas with non-strict generators have infinite
density along the boundary of the level set $C(u,v)=0$ (see e.g. Table 4.2 in
[44]). In these cases, the SPAR assumptions are not satisfied on GP margins.

The use of short-tailed margins and copulas whose support does not cover the
whole of $(0,1)^{2}$ has natural applications to some environmental variables,
such as winds, waves, temperatures, etc., discussed in Section 1.1. It many
cases it is reasonable to assume that an environmental variable has an upper
limit due to some physical constraints. For example, the maximum possible
water surface wave height at a given location is limited by the water depth.
Even if physical constraints are not fully understood, in many cases it is
reasonable to expect that values of environmental variables cannot go to
infinity. Moreover, some combinations of variables may also be ruled out by
physical constraints. Sticking with the example of waves, combinations of wave
heights, $H$, and wavelengths, $L$, that are physically possible are
constrained by the wave steepness, defined as the ratio $H/L$.viiiviiiviiiIn
deep water, the wavelength is proportional to the square of the wave period –
see Figure 2(a). When the steepness exceeds a certain value, the wave breaks,
limiting the height as a function of wavelength. Therefore, the range of
possible values of the random vector $(H,L)$ is bounded by a line of the form
$H=sL$, for some limiting steepness $s$. The analogous behaviour of the
density of the Clayton copula on GP margins with negative shape, illustrated
in Figure 15, is clearly evident, with the joint density bounded by the line
$x+y=-1/\alpha$. However, when modelling environmental variables which are
assumed to have bounded support, it may be more appropriate not to use
prescribed margins, and instead estimate the SPAR model on suitably scaled and
centred data. This is discussed further in the following section.

In summary, Example 14 shows that the independence copula does not have a SPAR
representation on GP margins with negative shape parameter, and illustrates
that there are constraints on the types of copula that have SPAR
representations on these margins. However, Examples 15 and 16 show that some
copulas whose support does not cover the whole of $(0,1)^{2}$ do have SPAR
representation on short-tailed margins.

##  6 Discussion and conclusions

This work has presented a new and general framework for modelling multivariate
extremes, based on an angular-radial representation of the density function.
The framework assumes a transformation of variables from Cartesian to polar
coordinates. In Section 3 it was shown that the choice of generalised polar
coordinates, introduced in Section 2, does not affect whether the SPAR model
assumptions are satisfied, but that certain coordinate systems lead to simpler
representations in which the angular and radial variables are asymptotically
independent. However, the coordinate systems for which this is true depends on
both the copula and margins. So, in general, it seems more appropriate to work
with polar coordinates defined in terms of the $L^{1}$ norm, which are
straightforward to work with in higher dimensions, and result in a simple
transformation between Cartesian and polar density functions.

For the theoretical examples considered in this work, the SPAR model
assumption (A1), that the conditional radial distribution converges to a GP
distribution was always satisfied. However, the more restrictive assumptions
(A2) and (A3), regarding continuity and finite values for the angular density
and GP parameter functions, were not always satisfied. For a given copula, the
choice of margin affects whether the SPAR model assumptions are satisfied. It
was shown that the use of Laplace margins leads to density functions where the
SPAR model assumptions are satisfied for many commonly-used copulas. Using
exponential margins leads to similar asymptotic representations in the non-
negative orthant of $\mathbb{R}^{d}$. However, in certain cases the angular
density can become unbounded on exponential margins, whereas it remains finite
on Laplace margins. So there is some advantage to using Laplace margins over
exponential margins, even when the interest is only in extremes in the non-
negative orthant of $\mathbb{R}^{d}$.

SPAR models on Laplace margins also have a useful link to the limit set of the
scaled sample cloud, when it exists. The scale parameter function of a SPAR
model on Laplace margins describes the radius of the boundary of the limit set
as a function of angle. However, it is important to note that the for some
copulas, the limit set is degenerate in some regions (i.e. the boundary has
zero radius from the origin at some angles). For the examples considered where
this was the case, the joint densities still have SPAR representations on
Laplace margins. While limit sets are of interest from a theoretical point of
view, estimation of the density in extreme regions (or the corresponding
probability of observations falling within an extreme sets) is usually of
greater practical importance. Characterising the density in extreme regions of
variable space is the main aim of the SPAR approach.

SPAR models on Laplace margins can represent a wide range of copulas. However,
for copulas with strong dependence in the upper tail, it was shown that the
joint exceedance region in the upper tail has a more natural representation on
Pareto margins. From a probabilistic perspective, SPAR representations for
these cases are equivalent to the representation derived by [33]. Similarly,
for copulas with $\kappa_{\mathbf{1}_{d}}>1$, the SPAR representation on
Pareto margins is equivalent to the Ledford-Tawn model [11, 12]. However, in
both cases $\kappa_{\mathbf{1}_{d}}=1$ and $\kappa_{\mathbf{1}_{d}}>1$, there
are copulas for which the SPAR representation on Pareto margins is only valid
in the region where all variables are large, and is not valid when one or more
variables are small. In contrast, in the examples presented, the use of
Laplace margins for such copulas leads to SPAR representations which are valid
for all angles.

With regard to how the SPAR framework relates to existing approaches, we can
summarise that the models of [33], [11] and [14] are all special cases of SPAR
representations, and that SPAR models can represent extremal properties of a
wider range of distributions than these models. Moreover, the angular-radial
decomposition of the joint density function, into an angular density and
conditional radial density, provides a simple intuitive motivation for our
model. Viewed in this way, multivariate extremes can be seen as a simple
extension to univariate extremes, with angular dependence. It is important to
note that the joint distribution function does not admit an analogous
decomposition in general, which highlights why it is important to begin by
considering the joint density function, rather than joint distribution
function.

The focus of the present work has been on probabilistic aspects of SPAR
representations, with particular consideration of the effects of the choice of
coordinate system and the choice of margins. The motivation is that these
theoretical considerations will inform an approach to inference, which will be
developed in further work. The general conclusion from the present work is
that working on Laplace margins in $L^{1}$ polar coordinates provides a
variable space in which the SPAR model assumptions are satisfied for a wide
range of copulas, with various tail dependence properties. As mentioned in the
introduction, the SPAR model can be viewed as a type of non-stationary peaks-
over-threshold (POT) model, with angle as covariate. There are many examples
of inference for non-stationary POT in the literature, e.g. [1, 2, 3, 4]. In
the examples considered in this work, it was found that when the SPAR
assumptions are satisfied, the angular densities are smooth functions of
angle, which a readily amenable to non-parametric estimation. However,
although the GP scale functions are continuous (assumption A2), they were not
always smooth functions of angle. This will have implications for their
estimation using semi-parametric means, which often assume smoothly-varying
functions. For these cases, a piecewise-linear model (e.g. [4]) may be more
appropriate.

For a SPAR model on Laplace margins with constant shape parameter
$\xi(\mathbf{w})=0$, the tail order in corner
$\mathbf{u}_{0}=(u_{0,1},...,u_{0,d})$ is linked to the GP scale parameter by
$\sigma(\mathbf{w}_{0})=2/\kappa_{\mathbf{u}_{0}}$, where
$2\mathbf{w}_{0}=\left((-1)^{u_{0,1}+1},...,(-1)^{u_{0,d}+1}\right)$. However,
the relations between the SPAR model parameters and tail dependence
coefficients have not been considered in the present work. The general
dependence properties of SPAR models can be derived from the marginal and
joint exceedance probabilities, which can be calculated from integrals of the
SPAR density over angle. Whilst this is simple in principle, the calculations
involved are lengthy, and require consideration of numerous separate cases.
These considerations are therefore deferred to future work.

Section 5 considered the effect of the margins on whether the SPAR assumptions
are satisfied. However, as discussed in Section 1.1, conducting the inference
on prescribed margins is not a requirement of the SPAR approach. When applying
the model to observational data, generally the marginal distribution is not
known. In this case, working on prescribed margins requires first estimating
the distribution for each margin and then applying a transformation. This will
introduce further uncertainty to the estimated joint distribution. An
alternative approach would be to apply the SPAR model to the observations
directly, after a suitable normalisation for scale and location, and estimate
both the extremal properties of the margins and copula in a single inference.
This type of approach may be applicable when it is reasonable to assume that
all variables have a finite upper and lower bounds. As discussed in Section
5.4, this is a reasonable assumption for most environmental applications. This
type of approach will be considered in future work, where the inference for
SPAR models is discussed in detail.

A further consideration for the application of the SPAR model to real
datasets, is where to define the origin of the polar coordinate system. When
working on prescribed margins, the natural choice is to define the polar
origin to coincide with the Cartesian origin. For distributions which have
support in all of $\mathbb{R}^{d}$, a finite shift in the origin has no effect
on the asymptotic properties of the model, but can result in a faster
convergence to the asymptotic form, as discussed in Example 11. However, on
short-tailed margins, the choice of origin can have an impact on the SPAR
model parameters, or even whether the model assumptions are satisfied. For
instance, suppose that random vector $\mathbf{X}\in\mathbb{R}^{d}$ has bounded
support, $\text{supp}(\mathbf{X})=G\subset\mathbb{R}^{d}$. The bounded set $G$
can be viewed as analogous to a limit set for random vectors with unbounded
support. Clearly, a minimum requirement for the SPAR representation when $G$
is bounded, is that $G$ is star-shaped with respect to the origin. The choice
of origin for cases where random vector $\mathbf{X}\in\mathbb{R}^{d}$ has
bounded support will be discussed in detail in future work.

## Acknowledgement

This work was funded by the EPSRC, United Kingdom Supergen Offshore Renewable
Energy Hub [EP/S000747/1] Flexible Fund project “Improved Models for
Multivariate Metocean Extremes (IMEX)”. The data shown in Figures 1 and 2 was
provided as part of the Benchmarking Exercise for Environmental Contours [45]
and is available from https://github.com/ec-benchmark-organizers. The generic
response curves shown in Figures 1(a) and 1(c) were reproduced from [46] and
[47], respectively. We would like to thank Callum Murphy-Barltrop for useful
comments on a draft of this paper.

##  Appendix A Proofs

### Proof of Proposition 3.1

Define $u=\cos_{p}(q)$ and $v=\sin_{p}(q)$. Then we can write

$\displaystyle\cos_{p}^{\prime}(q)$ $\displaystyle=\frac{du}{dq},$
$\displaystyle\sin_{p}^{\prime}(q)$
$\displaystyle=\frac{dv}{dq}=\frac{du}{dq}\frac{dv}{du}.$

Differentiating (6), the rate of change of pseudo-angle with $u$ is

$\left|\frac{dq}{du}\right|=\frac{4}{\mathcal{C}_{p}}\left(1+\left(\frac{dv}{du}\right)^{2}\right)^{1/2},$

where $\mathcal{C}_{p}$ is the circumference of the $L^{p}$ unit circle (see
Definition 8). Substituting these expressions into (10) gives

$J_{p}(q)=\frac{\mathcal{C}_{p}}{4}\left|u\frac{dv}{du}-v\right|\left(1+\left(\frac{dv}{du}\right)^{2}\right)^{-1/2}.$
(31)

To simplify the presentation, we just consider the case for the first
quadrant. Results for other quadrants follow by symmetry. For $L^{p}$ norms,
in the first quadrant we have $v=(1-u^{p})^{1/p}$. Therefore

$\frac{dv}{du}=-u^{p-1}\left(1-u^{p}\right)^{\tfrac{1}{p}-1}=-\left(\frac{u}{v}\right)^{p-1}.$

Substituting this gives

$\left|u\frac{dv}{du}-v\right|=\left|\frac{u^{p}}{v^{p-1}}+v\right|=v^{1-p}\left|u^{p}+v^{p}\right|=v^{1-p}.$

Then, using (31) we have

$J_{p}(q)=\frac{\mathcal{C}_{*}}{4}v^{1-p}\left(1+\left|\frac{u}{v}\right|^{2p-2}\right)^{-1/2}=\frac{\mathcal{C}_{*}}{4}\left(u^{2(p-1)}+v^{2(p-1)}\right)^{-1/2}.$

Substituting $u=\cos_{p}(q)$, $v=\sin_{p}(q)$ gives the first part of the
result. The specific forms of the Jacobian for $p=1$, 2 and $\infty$ are
straightforward to verify and follow directly from Example 1.

For the last part of the result, we again restrict attention to the first
quadrant. We write $J_{p}$ as a function of $u$, substituting
$v=(1-u^{p})^{1/p}$, and consider which values of $p$ lead to zero derivative.
We have

$\frac{dJ_{p}}{du}=2(p-1)\left[u^{2p-3}-u^{p-1}(1-u^{p})^{1-2/p}\right].$

For $p\in(1,2)\cup(2,\infty)$ we have $dJ_{p}/du=0$ implies $u^{p-2}=v^{p-2}$,
which only occurs when $u=v$, and completes the proof.

### Proof of Lemma 3.2

For a vector $\mathbf{x}\in\mathbb{R}^{2}\setminus\\{\mathbf{0}\\}$, define
$r_{a}=\mathcal{R}_{a}(x,y)$, $r_{b}=\mathcal{R}_{b}(x,y)$ and
$q=\mathcal{A}_{*}^{(-2,2]}((x,y)/\mathcal{R}_{*}(x,y))$. Denote
$\mathbf{w}=(\cos_{*}(q),\sin_{*}(q))$. Then we can write

$\mathbf{x}=r_{a}\frac{\mathbf{w}}{\mathcal{R}_{a}(\mathbf{w})}=r_{b}\frac{\mathbf{w}}{\mathcal{R}_{b}(\mathbf{w})}.$

Hence

$r_{a}=r_{b}\frac{\mathcal{R}_{a}(\mathbf{w})}{\mathcal{R}_{b}(\mathbf{w})}.$
(32)

Therefore
$dr_{b}/dr_{a}=\mathcal{R}_{b}(\mathbf{w})/\mathcal{R}_{a}(\mathbf{w})$, and
the result follows by substituting the Jacobian.

### Proof of Corollary 3.2.1

Denote the upper end point of $R|(Q=q)$ as $r_{F}(q)$, and denote
$\mathbf{w}=(\cos_{*}(q),\sin_{*}(q))$. From Lemma 3.2 we can use the change
of variables (32) and write

$\displaystyle\int_{0}^{r_{F}(q)}f_{R_{b},Q}(r_{b},q)dr_{b}$
$\displaystyle=\frac{\mathcal{R}_{a}(\mathbf{w})}{\mathcal{R}_{b}(\mathbf{w})}\int_{0}^{r_{F}(q)}f_{R_{a},Q}\left(r_{b}\frac{\mathcal{R}_{a}(\mathbf{w})}{\mathcal{R}_{b}(\mathbf{w})},q\right)dr_{b}=\int_{0}^{r_{F}(q)}f_{R_{a},Q}\left(r_{a},q\right)dr_{a}=f_{Q}(q).$

So the angular density $f_{Q}(q)$ remains unchanged by the change of gauge
function and satisfies (A3) from the assumptions of the corollary. The
relation between the conditional quantile functions $\mu_{a}$ and $\mu_{b}$
follows by noting that for given $\zeta$ and $q$, these must both refer to the
same point in Cartesian space, so must be related by (32). Similarly, the
conditional upper end points of the conditional radial distributions,
$r_{a,F}(q)$ and $r_{b,F}(q)$ must also be related by (32).

To demonstrate $f_{R_{b},Q}$ satisfies (A1) we have (writing $\mu_{a}$ and
$\mu_{b}$ for brevity, instead of $\mu_{a}(\zeta,q)$, $\mu_{b}(\zeta,q)$)

$\displaystyle\lim_{\zeta\to
1}\sup_{r_{b}\in[0,r_{b,F}(q)-\mu_{b}]}\left|\frac{\bar{F}_{R_{b}|Q}(\mu_{b}+r_{b}|q)}{\bar{F}_{R_{b}|Q}(\mu_{b}|q)}-\bar{F}_{GP}(r_{b};\xi(q),\sigma_{b}(\mu_{b},q))\right|$
$\displaystyle=$ $\displaystyle\lim_{\zeta\to
1}\sup_{r_{b}\in[0,r_{b,F}(q)-\mu_{b}]}\left|\frac{\int_{\mu_{b}+r_{b}}^{r_{F}(q)}f_{R_{b}|Q}(s|q)ds}{\int_{\mu_{b}}^{r_{F}(q)}f_{R_{b}|Q}(s|q)ds}-\left(1+\xi(q)\frac{r_{b}}{\sigma_{b}(\mu_{b},q)}\right)^{-1/\xi(q)}\right|$
$\displaystyle=$ $\displaystyle\lim_{\zeta\to
1}\sup_{r_{a}\in[0,r_{a,F}(q)-\mu_{a}]}\left|\frac{\int_{\mu_{a}+r_{a}}^{r_{F}(q)}f_{R_{a}|Q}(s|q)ds}{\int_{\mu_{a}}^{r_{F}(q)}f_{R_{a}|Q}(s|q)ds}-\left(1+\xi(q)\frac{r_{a}\frac{\mathcal{R}_{b}(\mathbf{w})}{\mathcal{R}_{a}(\mathbf{w})}}{\sigma_{b}(\mu_{b},q)}\right)^{-1/\xi(q)}\right|$
$\displaystyle=$ $\displaystyle\lim_{\zeta\to
1}\sup_{r_{a}\in[0,r_{a,F}(q)-\mu_{a}]}\left|\frac{\bar{F}_{R_{a}|Q}(\mu_{a}+r_{a}|q)}{\bar{F}_{R_{a}|Q}(\mu_{a}|q)}-\left(1+\xi(q)\frac{r_{a}}{\sigma_{a}(\mu_{a},q)}\right)^{-1/\xi(q)}\right|=0.$

Here we have tacitly assumed that $\xi(q)\neq 0$. For values of $q$ where
$\xi(q)=0$, the proof is identical to that above, substituting the appropriate
form of the GP survivor function. Regarding assumption (A2), about the
finiteness and continuity of the GP parameter functions, we note that the
ration $\mathcal{R}_{a}(\mathbf{w})/\mathcal{R}_{b}(\mathbf{w})$ is finite and
continuous, since it was stipulated that polar coordinate systems are defined
in terms of gauge functions for continuous surfaces (see Definition 9). Hence
if $\sigma_{a}$ satisfies (A2), so does $\sigma_{b}$.

### Proof of Theorem 3.3

Applying Corollary 3.2.1 to $f_{S,Q}$ shows that (i) implies (ii). For the
second part, first note that $\mathcal{R}_{\alpha}$ must define a gauge
function, since the set
$\\{\mathbf{x}\in\mathbb{R}^{2}:\mathbf{x}\leq\sigma(q)(\cos_{*}(q),\sin_{*}(q)),q\in
I\\}$ is star-shaped and has continuous boundary. Then, applying Corollary
3.2.1 to $f_{R,Q}$ with $S=\mathcal{R}_{\alpha}(R)$ shows that (ii) implies
(i).

### Proof of Lemma 3.4

The first part of the lemma is simply a statement of the general rule for
transformation of coordinate systems for probability density functions. Note
that the Jacobian is always positive, since any function $q_{a,b}(q)$ must be
monotonically increasing. The particular forms of $q_{1,2}$ and $q_{2,1}$
follow immediately from Example (1). For the Jacobians we just need to
consider the case $q\in[0,1]$. The Jacobians in other quadrants follow by
symmetry. For $q\in[0,1]$ we can write

$q_{1,2}(q)=\frac{\sin(\pi q/2)}{\cos(\pi q/2)+\sin(\pi q/2)}.$

Differentiating this expression gives the first part of the result. For the
second part, for $q\in[0,1]$ we can write
$q_{2,1}=(2/\pi)\mathrm{atan}(q/(1-q))$, so that

$\frac{dq_{2,1}(q)}{dq}=\frac{2}{\pi}(2q^{2}-2q+1)^{-1}=\frac{2}{\pi}\left(\cos_{1}^{2}(q)+\sin_{1}^{2}(q)\right)^{-1}.$

### Proof of Corollary 3.4.1

From Lemma 3.4, for $q\in[q_{b,a}(s),q_{b,a}(t)]$ we have

$\displaystyle f_{Q_{b}}(q)$
$\displaystyle=\int_{0}^{r_{F}(q)}f_{R,Q_{b}}(r,q)dr=\left|\frac{dq_{a,b}(q)}{dq}\right|\int_{0}^{r_{F}(q)}f_{R,Q_{a}}(r,q_{a,b}(q))dr=\left|\frac{dq_{a,b}(q)}{dq}\right|f_{Q_{a}}\left(q_{a,b}(q)\right),$

where $r_{F}(q)$ is the upper end point of $R|Q_{b}$. Since $f_{Q_{a}}$ is
continuous and finite for $q\in[s,t]$ and hence satisfies assumption (A3),
$f_{Q_{b}}(q)$ also satisfies assumption (A3). For the conditional radial
density, we have

$\displaystyle
f_{R|Q_{b}}(r|q)=\frac{f_{R,Q_{b}}(r,q)}{f_{Q_{b}}(q)}=\frac{f_{R,Q_{a}}(r,q_{a,b}(q))}{f_{Q_{a}}(q_{a,b}(q))}=f_{R|Q_{a}}(r|q_{a,b}(q)).$

Therefore, the conditional radial density $f_{R|Q_{b}}(r|q)$ satisfies the
SPAR assumptions (A1) and (A2) for the mapped angular range
$q\in[q_{b,a}(s),q_{b,a}(t)]$.

### Proof of Proposition 4.1

We shall by considering the relation between the ARE and ARL models for the
copula. The relation between the models for the copula density is derived in
the same way. Under the assumptions of the proposition, for
$\mathbf{w}=(1-w,w)$ and $w\in(0,1)$ we have

$C_{\mathbf{u}_{0}}(r^{1-w},r^{w})\sim\mathcal{L}_{\mathbf{u}_{0}}(r,\mathbf{w})r^{\Lambda_{\mathbf{u}_{0}}(\mathbf{w})},\quad
r\to 0^{+},$

where $\mathcal{L}_{\mathbf{u}_{0}}$ is slowly-varying in $r$ at $0^{+}$. We
wish to derive a representation of the form (17), for $\mathbf{z}=(1-z,z)$,
$z\in(0,1)$, given by

$C_{\mathbf{u}_{0}}(t\mathbf{z})\sim
B_{\mathbf{u}_{0}}(\mathbf{z})\tilde{\mathcal{L}}_{\mathbf{u}_{0}}(t)t^{\kappa_{\mathbf{u}_{0}}},\quad
t\to 0^{+},$

where $\tilde{\mathcal{L}}_{\mathbf{u}_{0}}\in RV_{0}(0^{+})$. Equating the
variables of each model, $r^{1-w}=t(1-z)$, $r^{w}=tz$, and solving for $r$ and
$w$ gives

$\displaystyle r$ $\displaystyle=t^{2}z(1-z),$ $\displaystyle w$
$\displaystyle=\frac{\log(tz)}{\log(t^{2}z(1-z))}=\frac{1}{2}+\frac{\log(z)-\log(1-z)}{2\log(t^{2}z(1-z))}.$

For fixed $z$, $w\to 1/2$ for $t\to 0^{+}$. Hence
$\mathcal{L}_{\mathbf{u}_{0}}(r,\mathbf{w})\sim\tilde{\mathcal{L}}_{\mathbf{u}_{0}}(t)$
as $t\to 0^{+}$ for some $\tilde{\mathcal{L}}_{\mathbf{u}_{0}}\in
RV_{0}(0^{+})$. From the assumptions of the proposition, we can expand
$\Lambda_{\mathbf{u}_{0}}(1-z,z)$ as a Taylor series about $z=1/2$:

$\displaystyle\Lambda_{\mathbf{u}_{0}}(1-z,z)=\Lambda_{\mathbf{u}_{0}}(\tfrac{1}{2},\tfrac{1}{2})+\beta(z-\tfrac{1}{2})+O(z-\tfrac{1}{2})^{2},$

where $\beta=[\tfrac{d}{dz}\Lambda_{\mathbf{u}_{0}}(1-z,z)]_{z=1/2}$. From the
discussion in Section 4.2, we have
$\Lambda_{\mathbf{u}_{0}}(\tfrac{1}{2},\tfrac{1}{2})=\tfrac{1}{2}\kappa_{\mathbf{u}_{0}}$.
Therefore, we have

$\displaystyle r^{\Lambda_{\mathbf{u}_{0}}(\mathbf{w})}$
$\displaystyle=\exp\left(\log(t^{2}z(1-z))\left[\frac{1}{2}\kappa_{\mathbf{u}_{0}}+\beta\frac{\log(z)-\log(1-z)}{2\log(t^{2}z(1-z))}+O(\log(t))^{-2}\right]\right)$
$\displaystyle=(t^{2}z(1-z))^{\tfrac{\kappa_{\mathbf{u}_{0}}}{2}}\left(\frac{z}{1-z}\right)^{\tfrac{\beta}{2}}O(1).$

This gives the required form of $B_{\mathbf{u}_{0}}(\mathbf{z})$, stated in
the proposition. The required form of $B_{\mathbf{u}_{0}}$ is obtained in the
same way. For the final claim, note that when $\kappa_{\mathbf{u}_{0}}<2$ and
$\beta=0$, the tail density function tends to infinity for $z\to 0^{+}$ and
$z\to 1^{-}$. Since $t^{\kappa_{\mathbf{u}_{0}}-2}\to\infty$ as $t\to 0^{+}$,
if the representation (18) was valid for $t\to 0^{+}$ and $z\to 0^{+}$ (or
$z\to 1^{-}$) simultaneously, then this would imply that copula density is
unbounded over a finite region, which is a contradiction, since the integral
of the density over $(u,v)\in[0,1]^{2}$ must be equal to one.

### Proof of Proposition 4.2

This proof follows a similar reasoning to that for Proposition 4.1. From the
assumptions of the proposition we have $c_{\mathbf{u}_{0}}(t(1-z),tz)\sim
b_{\mathbf{u}_{0}}(1-z,z)\mathcal{L}_{\mathbf{u}_{0}}(t)\,t^{\kappa_{\mathbf{u}_{0}}-2}$
for $t\to 0^{+}$ and $z\in[0,1]$, and also
$c_{\mathbf{u}_{0}}(r^{1-w},r^{w})\sim\mathcal{M}_{\mathbf{u}_{0}}(r;w)\,r^{\lambda_{\mathbf{u}_{0}}(\mathbf{w})-1}$
for $r\to 0^{+}$ and $w\in(0,1)$. Equating the variables of each model and
solving for $t$ and $z$ gives

$\displaystyle t$ $\displaystyle=r^{1-w}+r^{w}\sim
r^{\min(w,1-w)}\times\begin{cases}1,&w\in(0,1/2)\cup(1/2,1),\\\ 2,&w=1/2,\\\
\end{cases},\qquad r\to 0^{+},$ $\displaystyle z$
$\displaystyle=\frac{r^{w}}{r^{1-w}+r^{w}}\sim\begin{cases}1-r^{1-2w},&w<1/2,\\\
1/2,&w=1/2,\\\ r^{2w-1},&w>1/2,\end{cases}\qquad r\to 0^{+}.$

When $w=1/2$, we have

$\displaystyle
c_{\mathbf{u}_{0}}(r^{1/2},r^{1/2})=c_{\mathbf{u}_{0}}(t/2,t/2)\sim
b_{\mathbf{u}_{0}}(1/2,1/2)\mathcal{L}_{\mathbf{u}_{0}}(2r^{1/2})\,(2r)^{\tfrac{\kappa_{\mathbf{u}_{0}}}{2}-1}.$

Since $b_{\mathbf{u}_{0}}(1-z,z)\in RV_{\beta_{1}}(0^{+})$, for $w>1/2$ and
$r\to 0^{+}$ we have
$b_{\mathbf{u}_{0}}(1-z,z)=\mathcal{M}_{0}(r,w)r^{\beta_{1}(2w-1)}$ for some
function $\mathcal{M}_{0}$ that is slowly-varying in $r$ at $0^{+}$. Also note
that $\mathcal{L}_{\mathbf{u}_{0}}(t)$ is also slowly-varying in $r$. So,
putting these together, for $w>1/2$ we have

$\displaystyle
c_{\mathbf{u}_{0}}(r^{1-w},r^{w})=c_{\mathbf{u}_{0}}(t,(1-z),tz)\sim\mathcal{M}_{\mathbf{u}_{0}}(r;w)r^{\beta_{1}(2w-1)+(1-w)(\kappa_{\mathbf{u}_{0}}-2)},\quad
r\to 0^{+},$

for some function $\mathcal{M}_{\mathbf{u}_{0}}(r;w)$ that is slowly-varying
in $r$ at $0^{+}$. Similarly, for $w<1/2$, we have

$\displaystyle
c_{\mathbf{u}_{0}}(r^{1-w},r^{w})=c_{\mathbf{u}_{0}}(t,(1-z),tz)\sim\mathcal{M}_{\mathbf{u}_{0}}(r;w)r^{\beta_{2}(1-2w)+w(\kappa_{\mathbf{u}_{0}}-2)},\quad
r\to 0^{+}.$

Combining these results gives the desired form of $\lambda_{\mathbf{u}_{0}}$.

### Proof of Proposition 4.3

For
$\mathbf{w}\in\\{(w_{1},...,w_{d})\in\mathcal{U}_{1}:(-1)^{1+u_{d,j}}w_{j}>0,\,j=1,...,d\\}$,
we can write

$\displaystyle\delta_{L}(r,\mathbf{w})=c_{\mathbf{u}_{0}}(\tfrac{1}{2}\exp(-rw_{1}),...,\tfrac{1}{2}\exp(-rw_{d}))=c_{\mathbf{u}_{0}}(\exp(-tz_{1}),...,\exp(-tz_{d})).$

Equating marginal-scale and copula-scale coordinates gives for $j=1,...,d$,
gives

$\frac{1}{2}\exp(-rw_{j})=\exp(-tz_{j})\iff\log(2)+rw_{j}=tz_{j}.$

To solve for $t$ and $\mathbf{z}$, we assume $\mathbf{z}\in\mathcal{U}_{1}$
(this is always possible by rescaling), which gives

$\displaystyle t$ $\displaystyle=\sum_{j=1}^{d}tz_{j}=d\log(2)+r,$
$\displaystyle z_{j}$
$\displaystyle=\frac{\log(2)+rw_{j}}{d\log(2)+r}=w_{j}+O(r^{-1}).$

When $|w_{j}|=1/d$ for $j=1,...d$, we have $z_{j}=w_{j}$ for $j=1,...d$, and

$\displaystyle\delta_{L}(r,\mathbf{w})$
$\displaystyle\sim\mathcal{M}_{\mathbf{u}_{0}}(\exp(-t),\mathbf{z})\exp(-t(\lambda_{\mathbf{u}_{0}}(\mathbf{z})-1)),\qquad
r\to\infty$
$\displaystyle=\mathcal{M}_{\mathbf{u}_{0}}(2^{-d}\exp(-r),\mathbf{w})\exp(-(d\log(2)+r)(\lambda_{\mathbf{u}_{0}}(\mathbf{w})-1))$
$\displaystyle=\mathcal{M}_{\mathbf{u}_{0}}(2^{-d}\exp(-r),\mathbf{w})2^{-d(\lambda_{\mathbf{u}_{0}}(\mathbf{w})-1)}\exp(-r(\lambda_{\mathbf{u}_{0}}(\mathbf{w})-1)).$

For other values of
$\mathbf{w}\in\\{(w_{1},...,w_{d})\in\mathcal{U}_{1}:(-1)^{1+u_{d,j}}w_{j}>0,\,j=1,...,d\\}$,
since $\lambda_{\mathbf{u}_{0}}$ is continuous and has bounded partial
derivatives everywhere apart from the ray $\mathbf{z}=\mathbf{1}_{d}$, we have
$\lambda_{\mathbf{u}_{0}}(\mathbf{z})=\lambda_{\mathbf{u}_{0}}(\mathbf{w}+O(r^{-1}))=\lambda_{\mathbf{u}_{0}}(\mathbf{w})+O(r^{-1})$.
Therefore, for $r\to\infty$ we can write

$\displaystyle\delta_{L}(r,\mathbf{w})$
$\displaystyle\sim\mathcal{M}_{\mathbf{u}_{0}}(\exp(-t),\mathbf{z})\exp(-t(\lambda_{\mathbf{u}_{0}}(\mathbf{z})-1))$
$\displaystyle=\mathcal{M}_{\mathbf{u}_{0}}(2^{-d}\exp(-r),\mathbf{w}+O(r^{-1}))\exp(-(d\log(2)+r)(\lambda_{\mathbf{u}_{0}}(\mathbf{w})-1+O(r^{-1})))$
$\displaystyle=\mathcal{M}_{\mathbf{u}_{0}}(2^{-d}\exp(-r),\mathbf{w}+O(r^{-1}))O(1)\exp(-r(\lambda_{\mathbf{u}_{0}}(\mathbf{w})-1)).$

The terms outside the exponential function are slowly-varying in $\exp(r)$ at
$\infty$, which completes the proof.

### Proof of Proposition 4.4

For this case we write

$\displaystyle\delta_{GP}(r,\mathbf{w})=c_{\mathbf{1}_{d}}((1+\xi_{m}rw_{1})^{-1/\xi_{m}},...,(1+\xi_{m}rw_{d})^{-1/\xi_{m}})=c_{\mathbf{1}_{d}}(t\mathbf{z}).$

We can relate the copula-scale angular-radial coordinates $(t,\mathbf{z})$
with the marginal-scale angular-radial coordinates $(r,\mathbf{w})$ as
follows. If we assume $\mathbf{z}\in\mathcal{U}_{1}$ and denote
$s_{\mathbf{w}}=\xi_{m}^{-1/\xi_{m}}\sum_{j=1}^{d}w_{j}^{-1/\xi_{m}}$ then

$\displaystyle t$
$\displaystyle=\sum_{j=1}^{d}tz_{j}=\sum_{j=1}^{d}(1+\xi_{m}rw_{j})^{-1/\xi_{m}}\sim
r^{-1/\xi_{m}}s_{\mathbf{w}},\quad r\to\infty$ $\displaystyle z_{j}$
$\displaystyle=t^{-1}(1+\xi_{m}rw_{j})^{-1/\xi_{m}}\sim
w_{j}^{-1/\xi_{m}}s_{\mathbf{w}}^{-1},\quad r\to\infty.$

Therefore $\mathbf{z}$ is asymptotically constant along rays of constant
$\mathbf{w}$. Substituting this into the assumed asymptotic form of the copula
gives

$\delta_{GP}(r,\mathbf{w})\sim
s_{\mathbf{w}}^{\kappa_{\mathbf{1}_{d}}-d}b_{\mathbf{1}_{d}}\left(s_{\mathbf{w}}^{-1}\mathbf{w}^{-1/\xi_{m}}\right)\mathcal{L}_{\mathbf{1}_{d}}\left(r^{-1/\xi_{m}}\right)r^{\tfrac{d-\kappa_{\mathbf{1}_{d}}}{\xi_{m}}},\quad
r\to\infty.$

Above we have used the fact that since $\mathcal{L}_{\mathbf{1}_{d}}$ is
slowly-varying at $0^{+}$,
$\mathcal{L}_{\mathbf{1}_{d}}\left(r^{-1/\xi_{m}}s_{\mathbf{w}}\right)\sim\mathcal{L}_{\mathbf{1}_{d}}\left(r^{-1/\xi_{m}}\right)$
as $r\to\infty$.

### Proof of Proposition 5.1

To simplify the presentation, we use the notation
$\mathbf{w}=(\cos_{1}(q),\sin_{1}(q))$ and
$\mathbf{u}=(F_{*}(r\cos_{1}(q)),F_{*}(r\sin_{1}(q)))$ throughout this proof.
The general strategy for proving these results is as follows. We have

$f_{Q_{*}}(q)=\int_{0}^{r_{F}(q)}f_{R_{*},Q_{*}}(r,q)dr,$

where $r_{F}(q)$ is the upper end point of $R_{*}|(Q_{*}=q)$. We need to
examine the behaviour of the integrand $f_{R_{*},Q_{*}}(r,q)$ for
$r\in[0,r_{F}(q)]$. First, we write the joint angular-radial density as

$\displaystyle f_{R_{*},Q_{*}}(r,q)$
$\displaystyle=r\,m_{*}(r,\mathbf{w})c(\mathbf{u})$
$\displaystyle=r\,m_{*}(r,\mathbf{w})\delta_{*}(r,\mathbf{w}).$

To examine the behaviour of $f_{R_{*},Q_{*}}$ for $\mathbf{u}$ close to corner
$(u_{0},v_{0})$, we write
$\delta_{*}(r,\mathbf{w})=c_{(u_{0},v_{0})}(t(1-z),tz)$, where

$\displaystyle t(1-z)$ $\displaystyle=u_{0}+(-1)^{u_{0}}F_{*}(r\cos_{1}(q)),$
$\displaystyle tz$ $\displaystyle=v_{0}+(-1)^{v_{0}}F_{*}(r\sin_{1}(q))).$

We then find asymptotic relations between copula-scale polar coordinates
$(t,z)$ and marginal-scale polar coordinates $(r,q)$, so that the assumed
asymptotic ARL form (18) can be applied. To check whether the integral is
finite, we use the assumed regular variation property, together with
Karamata’s theorem (see e.g. [28]) which states that

  1. (i)

if $\alpha>-1$ then $g\in RV_{\alpha}(0^{+})$ implies
$G(x)=\int_{0}^{x}g(t)dt\in RV_{\alpha+1}(0^{+})$ and is finite;

  2. (ii)

if $\alpha<-1$ then $g\in RV_{\alpha}(\infty)$ implies
$G(x)=\int_{x}^{\infty}g(t)dt\in RV_{\alpha+1}(\infty)$ and is finite.

(a) For GP margins, we have $q\in[0,1]$ and hence $\mathbf{w}=(1-q,q)$. The
rays originate in the lower left corner of the copula. Write

$\displaystyle f_{R_{GP},Q_{GP}}(r,q)$
$\displaystyle=r\,[(1+\xi_{m}r(1-q))(1+\xi_{m}rq)]^{-1-\tfrac{1}{\xi_{m}}}c_{(0,0)}(t(1-z),tz),$

where

$\displaystyle t(1-z)$
$\displaystyle=1-(1+\xi_{m}r(1-q))^{-\tfrac{1}{\xi_{m}}}$ $\displaystyle tz$
$\displaystyle=1-(1+\xi_{m}rq)^{-\tfrac{1}{\xi_{m}}}.$

Solving for $t$ and $w$ and expanding as Taylor series about $r=0$ gives
$t\sim r$ and $z\sim q$ for $r\to 0^{+}$. Therefore, we have

$\displaystyle f_{R_{GP},Q_{GP}}(r,q)$ $\displaystyle\sim
b_{(0,0)}(1-q,q)\mathcal{L}_{(0,0)}(r)\,r^{\kappa_{(0,0)}-1},\quad r\to
0^{+}.$

Therefore, the angular radial density is bounded as $r\to 0^{+}$ since either
$\kappa_{(0,0)}=1$ and $\lim_{r\to
0^{+}}\mathcal{L}_{(0,0)}(r)=\Upsilon_{(0,0)}\leq 1$, or $\kappa_{(0,0)}>1$.

For the behaviour as $r\to\infty$, we need to consider three cases,
$\xi_{m}<0$, $\xi_{m}=0$ and $\xi_{m}>0$. When $\xi_{m}<0$, we have $r\leq
r_{F}(q)=\min\left(-(\xi_{m}(1-q))^{-1},-(\xi_{m}q)^{-1}\right)$. When $q<1/2$
the path terminates on the right hand edge of the copula and when $q>1/2$ the
path terminates on the upper edge. As the copula density is bounded away from
the corners, $f_{R_{GP},Q_{GP}}$ is also bounded for $r\in[0,r_{F}(q)]$.
Therefore $f_{Q_{GP}}(q)$ is also finite, since the integral is over a finite
range. For $q=1/2$, substitute $r=r_{F}(q)-s=-2/\xi_{m}-s$, so that

$\displaystyle f_{R_{GP},Q_{GP}}(r,1/2)$
$\displaystyle=\left(-\frac{2}{\xi_{m}}-s\right)\,(-\xi_{m}s/2)^{-2-\tfrac{2}{\xi_{m}}}c_{(1,1)}(t/2,t/2),$

where $t=2(-\xi_{m}s/2)^{-\tfrac{1}{\xi_{m}}}$. For $s\to 0^{+}$ we have

$\displaystyle f_{R_{GP},Q_{GP}}(r,1/2)$
$\displaystyle\sim-\frac{2}{\xi_{m}}\,(-\xi_{m}s/2)^{-2-\tfrac{2}{\xi_{m}}}b_{(1,1)}(1/2)\mathcal{L}_{(1,1)}(t)t^{\kappa_{(1,1)}-2}$
$\displaystyle\propto\mathcal{L}_{(1,1)}\left(s^{-\tfrac{1}{\xi_{m}}}\right)s^{-2-\tfrac{\kappa_{(1,1)}}{\xi_{m}}}.$

In the second line we have used the fact that since $\mathcal{L}_{(1,1)}$ is
slowly-varying at $0^{+}$, for all $a>0$,
$\mathcal{L}_{(1,1)}(at)\sim\mathcal{L}_{(1,1)}(t)$ as $t\to 0^{+}$. So
$f_{R_{GP},Q_{GP}}(r,1/2)$ is in $RV_{\alpha}(0^{+})$, with
$\alpha=-2-(\kappa_{(1,1)}/\xi_{m})$. Applying Karamata’s theorem, the
integral is finite when $\alpha>-1$ or $\xi_{m}>-\kappa_{(1,1)}$.

When $\xi_{m}>0$ we have

$\displaystyle f_{R_{GP},Q_{GP}}(r,q)$
$\displaystyle=r\,[(1+\xi_{m}r(1-q))(1+\xi_{m}rq)]^{-1-\tfrac{1}{\xi_{m}}}c_{(1,1)}(t(1-z),tz),$

where

$\displaystyle t(1-z)$ $\displaystyle=(1+\xi_{m}r(1-q))^{-\tfrac{1}{\xi_{m}}}$
$\displaystyle tz$ $\displaystyle=(1+\xi_{m}rq)^{-\tfrac{1}{\xi_{m}}}.$

Solving for $t$ and $z$ gives for $r\to\infty$

$\displaystyle t$ $\displaystyle\sim
r^{-\tfrac{1}{\xi_{m}}}\left[(\xi_{m}(1-q))^{-\tfrac{1}{\xi_{m}}}+(\xi_{m}q)^{-\tfrac{1}{\xi_{m}}}\right],$
$\displaystyle z$
$\displaystyle\sim\frac{q^{-\tfrac{1}{\xi_{m}}}}{(1-q)^{-\tfrac{1}{\xi_{m}}}+q^{-\tfrac{1}{\xi_{m}}}}.$

Therefore, rays of constant $q$ asymptote to rays of constant $z$ on the
copula scale. So for the joint density, we get

$\displaystyle f_{R_{GP},Q_{GP}}(r,q)$
$\displaystyle\sim(\xi_{m}^{2}q(1-q))^{-1-\tfrac{1}{\xi_{m}}}b_{(1,1)}(z)\left[(\xi_{m}(1-q))^{-\tfrac{1}{\xi_{m}}}+(\xi_{m}q)^{-\tfrac{1}{\xi_{m}}}\right]^{\kappa_{(1,1)}-2}\mathcal{L}_{(1,1)}(t)r^{-1-\tfrac{\kappa_{(1,1)}}{\xi_{m}}}.$

This is in $RV_{\alpha}(\infty)$ with $\alpha=-1-(\kappa_{(1,1)}/\xi_{m})<-1$.
Hence, by Karamata’s theorem, this integral is finite for any $\xi_{m}>0$.

Finally, in the case that $\xi_{m}=0$, we have

$\displaystyle f_{R_{GP},Q_{GP}}(r,q)$
$\displaystyle=r\exp(-r)c_{(1,1)}(t(1-z),tz),$

where $t(1-z)=\exp(-r(1-q))$ and $tz=\exp(-rq)$, so that

$\displaystyle t$ $\displaystyle=\exp(-r(1-q))+\exp(-rq),$ $\displaystyle z$
$\displaystyle=\frac{\exp(-rq)}{\exp(-r(1-q))+\exp(-rq)}.$

Note that for $q<1/2$, $z\sim\exp(-r(2q-1))$ as $r\to\infty$. In some cases,
the assumed asymptotic for of the copula is not valid for $t,w\to 0^{+}$
simultaneously. So in these cases, we note that $tc_{(1,1)}(t(1-z),tz)$ is
bounded for $t\to 0^{+}$ for all $z\in[0,1]$. (If $tc_{(1,1)}(t(1-z),tz)$ is
not bounded for $z\to 0^{+}$, then this contradicts the fact that
$\int_{0}^{1}c_{(1,1)}(t,0)dt=1$). Since $tc_{(1,1)}(t(1-z),tz)$ is bounded,
there exists $\beta\in(0,\infty)$ and $\tau>0$ such that
$tc_{(1,1)}(t(1-z),tz)<\beta$ for $t<\tau$. Then for $t<\tau$

$\displaystyle f_{R_{GP},Q_{GP}}(r,q)$ $\displaystyle\leq
r\exp(-r)\frac{\beta}{t}$
$\displaystyle=r\exp(-r)\frac{\beta}{\exp(-r(1-q))+\exp(-rq)}$
$\displaystyle\sim\beta
r\begin{cases}\exp(-r(1-\min(q,1-q))),&q\in(0,1/2)\cup(1/2,1),\\\
2\exp(-r/2),&q=1/2.\end{cases}$

So by the dominated convergence theorem, $f_{Q_{GP}}(q)$ is finite for
$\xi_{m}=0$, which completes the proof for part (a).

(b) For GP margins, as $q\to 0^{+}$ the paths $\mathbf{u}$ through the copula
pass closer to the lower right corner of the copula (see Figure 10). To
examine the behaviour of $f_{R_{GP},Q_{GP}}$ for $\mathbf{u}$ close to corner
$(1,0)$, we need to determine the range of $r$ for which
$\mathbf{u}=(F_{GP}(r(1-q)),F_{GP}(rq))\to(1^{-},0^{+})$ as $q\to 0^{+}$. For
part (i), we treat the cases $\xi_{m}=0$ and $\xi_{m}>0$ separately. For the
case of exponential margins we have

$\displaystyle f_{R_{E},Q_{E}}(r,q)=r\exp(-r)c_{(1,0)}(t(1-z),tz),$

where $t(1-z)=\exp(-r(1-q))$ and $tz=1-\exp(-rq)$. Solving for $t$ and $z$
gives

$\displaystyle t$ $\displaystyle=\exp(-r(1-q))+1-\exp(-rq),$ $\displaystyle z$
$\displaystyle=\frac{1-\exp(-rq)}{\exp(-r(1-q))+1-\exp(-rq)}.$

Consider the behaviour of the joint density in the range $r\in
J=[-\tfrac{1}{2}\log(q),-2\log(q)]$. For $q\to 0^{+}$, the lower limit of $J$
tends to infinity, but $rq\to 0$ for all $r\in J$. Therefore, for $r\in J$ and
$q\to 0^{+}$ we have

$\displaystyle t$ $\displaystyle\sim\exp(-r)+rq,$ $\displaystyle z$
$\displaystyle\sim\frac{rq}{\exp(-r)+rq}.$

For $r\in J$ and $q\to 0^{+}$, when $rq=\exp(-r)$, we have $z\to 1/2$. The
solution of $rq=\exp(-r)$ is $r=W_{0}(1/q)$, where $W_{0}$ is the zeroth
branch of the Lambert W function (see e.g. [48, §4.13]). Note that
$W_{0}(1/q)\sim-\log(q)-\log(-\log(q))$ as $q\to 0^{+}$. Define
$s=r-W_{0}(1/q)$, then for $r\in J$ and $q\to 0^{+}$ we have

$\displaystyle t$ $\displaystyle\sim\exp(-s-W_{0}(1/q))+(s+W_{0}(1/q))q$
$\displaystyle=(s+W_{0}(1/q))q(e^{-s}+1)$ $\displaystyle\sim
W_{0}(1/q)q(e^{-s}+1)$ $\displaystyle z$ $\displaystyle\sim(e^{-s}+1)^{-1}.$

Substituting this into the assumed form of the copula gives for $q\to 0^{+}$
and $r\in J$,

$\displaystyle
c_{(1,0)}(t(1-z),tz)\sim\Upsilon_{(1,0)}b_{(1,0)}\left(\frac{e^{-s}}{e^{-s}+1},\frac{1}{e^{-s}+1}\right)\left[W_{0}(1/q)q(e^{-s}+1)\right]^{-1},$

and

$\displaystyle f_{R_{E},Q_{E}}(r,q)$
$\displaystyle=(W_{0}(1/q)+s)W_{0}(1/q)q\exp(-s)c_{(1,0)}(t(1-z),tz)$
$\displaystyle\sim\Upsilon_{(1,0)}\frac{\exp(-s)}{\exp(-s)+1}b_{(1,0)}\left(\frac{e^{-s}}{e^{-s}+1},\frac{1}{e^{-s}+1}\right)W_{0}(1/q),$

Therefore, $f_{R_{E},Q_{E}}$ is unbounded for $r\in J$ as $q\to 0^{+}$, and
hence $f_{Q_{E}}(q)\to\infty$ as $q\to 0^{+}$.

For the case $\xi_{m}>0$ we take a similar approach to above and consider the
behaviour of the joint density for $r$ in the interval
$J=[-\tfrac{1}{2}\log(q),-2\log(q)]$. In this case we have

$\displaystyle
f_{R_{GP},Q_{GP}}(r,q)=r(1+\xi_{m}r(1-q))^{-1-\tfrac{1}{\xi_{m}}}(1+\xi_{m}rq)^{-1-\tfrac{1}{\xi_{m}}}c_{(1,0)}(t(1-z),tz),$

where $t(1-z)=(1+\xi_{m}r(1-q))^{-\tfrac{1}{\xi_{m}}}$ and
$tz=1-(1+\xi_{m}rq)^{-\tfrac{1}{\xi_{m}}}$. For $r\in J$ and $q\to 0^{+}$ we
have

$\displaystyle t(1-z)$ $\displaystyle\sim(\xi_{m}r)^{-\tfrac{1}{\xi_{m}}},$
$\displaystyle tz$ $\displaystyle\sim rq.$

Solving for $t$ and $z$, gives for $r\in J$ and $q\to 0^{+}$,

$\displaystyle t$ $\displaystyle\sim(\xi_{m}r)^{-\tfrac{1}{\xi_{m}}}+rq,$
$\displaystyle z$
$\displaystyle\sim\frac{rq}{(\xi_{m}r)^{-\tfrac{1}{\xi_{m}}}+rq}.$

Note that $z\to 1/2$ when $(\xi_{m}r)^{-\tfrac{1}{\xi_{m}}}=rq$ for $r\in J$
and $q\to 0^{+}$. If we substitute
$r=sq^{-1}(q/\xi_{m})^{\tfrac{1}{\xi_{m}+1}}$ then for $r\in J$ and $q\to
0^{+}$,

$\displaystyle t$
$\displaystyle\sim\left(\frac{q}{\xi_{m}}\right)^{\tfrac{1}{\xi_{m}+1}}\left[s^{-\tfrac{1}{\xi_{m}}}+s\right],$
$\displaystyle z$ $\displaystyle\sim\frac{s}{s^{-\tfrac{1}{\xi_{m}}}+s}.$

Substituting this for the joint density, and recalling that for $r\in J$ and
$q\to 0^{+}$ we have $r\to\infty$ and $rq\to 0^{+}$, so

$\displaystyle f_{R_{GP},Q_{GP}}(r,q)$
$\displaystyle=r(1+\xi_{m}r(1-q))^{-1-\tfrac{1}{\xi_{m}}}(1+\xi_{m}rq)^{-1-\tfrac{1}{\xi_{m}}}c_{(1,0)}(t(1-z),tz)$
$\displaystyle\sim\xi_{m}^{-1-\tfrac{1}{\xi_{m}}}r^{-\tfrac{1}{\xi_{m}}}c_{(1,0)}(t(1-z),tz)$
$\displaystyle\propto
q^{\tfrac{1}{\xi_{m}+1}}s^{-\tfrac{1}{\xi_{m}}}c_{(1,0)}(t(1-z),tz)$
$\displaystyle\sim
q^{\tfrac{1}{\xi_{m}+1}}s^{-\tfrac{1}{\xi_{m}}}\Upsilon_{(1,0)}b_{(1,0)}(1-z,z)t^{-1}$
$\displaystyle\propto
b_{(1,0)}(1-z,z)\frac{s^{-\tfrac{1}{\xi_{m}}}}{s^{-\tfrac{1}{\xi_{m}}}+s}.$

Therefore, for $r\in J$ and $q\to 0^{+}$ the joint density tends to a fixed
form in terms of $s$. However, since $dr/ds\propto
q^{-\tfrac{\xi_{m}}{\xi_{m}+1}}$, the integral of the joint density over $r$
diverges as $q\to 0^{+}$, which completes the proof for part (i).

For (ii) we proceed in the same way as for part (i), but using a different
change of variables. As with part (a), we substitute
$r=r_{F}(q)-s=a(1-s)/(1-q)$, where $a=-1/\xi_{m}$, to get $t(1-z)=s^{a}$ and
$tz=a(1-s)q+O(q^{2})$. Therefore, for $q\to 0^{+}$ we have

$\displaystyle t$ $\displaystyle\sim s^{a}+aq(1-s),$ $\displaystyle z$
$\displaystyle\sim\frac{aq(1-s)}{s^{a}+aq(1-s)}.$

As before, we are interested in the behaviour of the joint density around
$z=1/2$, which occurs when $aq=s^{a}/(1-s)$. The value $s^{a}/(1-s)$ is
monotonically increasing in $s$ for $s\in[0,1)$, so when $q$ is small, the
solution of $aq=s^{a}/(1-s)$ is approximately $s=(aq)^{1/a}$. Therefore, we
substitute $s=m(aq)^{\tfrac{1}{a}}$, with $m\in[0,(aq)^{-\tfrac{1}{a}}]$. As
we are interested in the behaviour of the joint density for $\mathbf{u}$ close
to $(1,0)$, we restrict our interest to $m\in J=[0,-\log(q)]$, so that the
upper end point of this interval tends to infinity as $q\to 0^{+}$, but
$mq^{b}\to 0^{+}$ for any $b>0$ and $m\in J$. So for $q\to 0^{+}$ and $m\in J$
we have $t\sim aq\left[m^{a}+1\right]$ and $z\sim(m^{a}+1)^{-1}$. We now write
the joint density as

$\displaystyle f_{R_{GP},Q_{GP}}(r,q)$
$\displaystyle=r(1+\xi_{m}r(1-q))^{-1-\tfrac{1}{\xi_{m}}}(1+\xi_{m}rq)^{-1-\tfrac{1}{\xi_{m}}}c_{(1,0)}(t(1-z),tz)$
$\displaystyle=a\frac{1-s}{1-q}\left[\left(1-q\frac{1-s}{1-q}\right)s\right]^{a-1}c_{(1,0)}(t(1-z),tz)$
$\displaystyle\sim as^{a-1}c_{(1,0)}(t(1-z),tz),\qquad q\to 0^{+},m\in J$
$\displaystyle=am^{a-1}(aq)^{\tfrac{a-1}{a}}c_{(1,0)}(t(1-z),tz)$
$\displaystyle\sim
am^{a-1}(aq)^{\tfrac{a-1}{a}}\Upsilon_{(1,0)}b_{(1,0)}(1-z,z)t^{-1}$
$\displaystyle\sim
a(aq)^{-\tfrac{1}{a}}\Upsilon_{(1,0)}b_{(1,0)}(1-z,z)\frac{m^{a-1}}{1+m^{a}}.$

Noting that $ds/dm=(aq)^{\tfrac{1}{a}}$, the integral of
$f_{R_{GP},Q_{GP}}(r,q)$ over $r$ converges to a fixed form in terms of $m$
for $\qquad q\to 0^{+}$ and $m\in J$. For $m\to 0^{+}$, $z\sim 1-m^{a}$, under
the assumptions above,

$\displaystyle f_{R_{GP},Q_{GP}}(r,q)$ $\displaystyle\sim
a(aq)^{-\tfrac{1}{a}}\Upsilon_{(1,0)}\mathcal{M}_{1}(1-m^{a})m^{a(\beta_{1}+1)-1},$

for some $\mathcal{M}_{1}(z)\in RV_{0}(1^{-})$. The integral
$\int_{0}^{m_{0}}f_{R_{GP},Q_{GP}}(r,q)dm$ for small $m_{0}$ has an integrand
that is independent of $q$ and is regularly-varying at $0^{+}$ with index
$\alpha=a(\beta_{1}+1)-1>-1$. Hence by Karamata’s theorem, this part of the
integral is finite. For the upper limit of the interval $J$, $m=-\log(q)$ and
$z\sim m^{-a}$ as $q\to 0^{+}$. So we can write

$\displaystyle f_{R_{GP},Q_{GP}}(r,q)$ $\displaystyle\sim
a(aq)^{-\tfrac{1}{a}}\Upsilon_{(1,0)}\mathcal{M}_{0}(m^{-a})m^{a\beta_{0}-1},$

for some $\mathcal{M}_{0}(z)\in RV_{0}(0^{+})$. As above, Karamata’s theorem
implies that the integral is finite as $q\to 0^{+}$ and $m\to\infty$. For the
remaining part of the integral, we note that $c$ is finite away from the
corners, so $f_{R_{GP},Q_{GP}}(r,q)$ is also finite and the range of the
integral is also finite. Hence $f_{Q_{GP}}(q)$ is finite as $q\to 0^{+}$.

(c) For symmetric margins, the rays of constant angle on the copula scale,
emanate from the centre of the copula $\mathbf{u}=(0.5,0.5)$ and pass close to
at most one corner of the copula – see Figure 10. Since the copula is assumed
to be continuous and bounded away from the corners, we need only consider the
behaviour of the joint density for $\mathbf{u}$ close to a corner of the
copula. The same steps used in part (a) then show that the density is
convergent when $q\notin\mathbb{Z}$, and the ray terminates in a corner. For
the cases where $q\in\mathbb{Z}$, the ray terminates on the edge of the
copula. From the assumed form of the copula, there is a $\beta>0$ such that
$c(\mathbf{u})\leq\beta$ for all values of $\mathbf{u}$ along the path and
hence $f_{R_{SGP},Q_{SGP}}\leq\beta\,r\,m_{SGP}(r,\mathbf{w})$. When
$\xi_{m}<0$, the range of the integral is finite and the joint density
$f_{R_{SGP},Q_{SGP}}$ is finite, so the integral is finite. When $\xi_{m}=0$,
we have $f_{R_{SGP},Q_{SGP}}\leq\tfrac{1}{4}\beta\,r\,\exp(-r)$, which has
finite integral over $r\in[0,\infty)$. Finally, when $\xi_{m}>0$ we have
$f_{Q_{SGP}}\leq\tfrac{1}{4}\beta\int_{0}^{\infty}r(1+\xi_{m}r)^{-1-1/\xi_{m}}dr$.
This integral is finite when $\xi_{m}<1$, which completes this part of the
proof.

(d) For simplicity, consider the case $q=0$ – other cases are treated
identically. In this case, the path terminates at $\mathbf{u}=(1,1/2)$. Since
$c$ is positive and finite on the edges $c(1,1/2)=\beta$ for some
$\beta\in(0,\infty)$. Since $c$ is continuous, for any $\epsilon\in(0,\beta)$
there exists $\Delta>0$ such that $c(1-\Delta,1/2)\geq\beta-\epsilon$.
Therefore for $\tfrac{1}{2}(1+\xi_{m}r)^{-1/\xi_{m}}<\Delta$, we have
$f_{R_{SGP},Q_{SGP}}(r,0)\geq\tfrac{1}{4}(\beta-\epsilon)r(1+\xi_{m}r)^{-1-1/\xi_{m}}$.
Define $r_{0}=((2\Delta)^{-\xi_{m}}-1)/\xi_{m}$. Then

$\displaystyle
f_{Q_{SGP}}(0)>\int_{r_{0}}^{\infty}f_{R_{SGP},Q_{SGP}}(r,0)dr\geq\frac{1}{4}(\beta-\epsilon)\int_{r_{0}}^{\infty}r(1+\xi_{m}r)^{-1-1/\xi_{m}}dr.$

The integral on the RHS is infinite, so since this inequality holds for any
$\epsilon>0$, $f_{Q_{SGP}}(0)$ is also infinite.

(e) For this case, we equate the copula-scale coordinate with the marginal-
scale coordinate, to give
$t=(1+\xi_{m}r)^{-1/\xi_{m}}\sim(\xi_{m}r)^{-1/\xi_{m}}$ for $r\to\infty$.
Then, since $c(1-t,1/2)\in RV_{\alpha}(0^{+})$, we have
$c(1-t,1/2)=\mathcal{L}(t)t^{\alpha}$ as $t\to 0^{+}$, for some function
$\mathcal{L}\in RV_{0}(0^{+})$. Therefore

$\displaystyle f_{R_{SGP},Q_{SGP}}(r,0)$
$\displaystyle=\frac{1}{4}r(1+\xi_{m}r)^{-1-\tfrac{1}{\xi_{m}}}\mathcal{L}\left(r^{-\tfrac{1}{\xi_{m}}}\right)(\xi_{m}r)^{-\tfrac{\alpha}{\xi_{m}}}$
$\displaystyle\sim\frac{1}{4}\xi_{m}^{-1-\tfrac{\alpha+1}{\xi_{m}}}\mathcal{L}\left(r^{-\tfrac{1}{\xi_{m}}}\right)r^{-\tfrac{\alpha+1}{\xi_{m}}},\quad
r\to\infty,$

Therefore $f_{R_{SGP},Q_{SGP}}(r,0)\in RV_{\beta}(\infty)$ with
$\beta=-(\alpha+1)/\xi_{m}$. Therefore, by Karamata’s theorem,
$f_{Q_{SGP}}(0)$ is finite when $\xi_{m}<\alpha+1$.

(f) For this case the joint density for $q=1/2$ is

$\displaystyle f_{R_{P},Q_{P}}(r,1/2)$
$\displaystyle=r(r/2)^{-4}c(t/2,t/2),\quad r\geq 2$

where $t/2=1-(2/r)$. Substitute, $r=2+s$ for $s\geq 0$, then $t\sim s$ for
$s\to 0^{+}$. Then we have

$\displaystyle f_{R_{P},Q_{P}}(2+s,1/2)$
$\displaystyle=(2+s)((2+s)/2)^{-4}c(t/2,t/2)$ $\displaystyle\sim
2\Upsilon_{(0,0)}b_{(0,0)}(1/2)s^{-1},\quad s\to 0^{+}.$

The integral of this function from 0 to any $s_{0}>0$ is infinite, and hence
$f_{Q_{P}}(1/2)$ is also infinite.

### Proof of Proposition 5.2

We have two cases to consider, depending on whether the path
$\mathbf{u}=(F_{L}(rw_{1}),...,F_{L}(rw_{d}))$ terminates in a corner of the
copula or not, as $r\to\infty$. First, assume that $w_{j}\neq 0$ for
$j=1,...,d$, so that the path $\mathbf{u}$ terminates in the corner
$\mathbf{u}_{0}=(H(w_{1}),...,H(w_{d}))$ as $r\to\infty$, where $H$ is the
Heaviside step function, $H(x)=1$ for $x\geq 0$ and $H(x)=0$ for $x<0$. By
Proposition 4.3, there exists a function
$\mathcal{M}_{\mathbf{u_{0}}}(t,\mathbf{w})$ that is slowly-varying in $t$ at
$0^{+}$, such that

$c_{\mathbf{u}_{0}}(\exp(-t\mathbf{z}))\sim\mathcal{M}_{\mathbf{u_{0}}}(\exp(-r),\mathbf{w})\exp(-r(\lambda_{\mathbf{u}_{0}}(\mathbf{w})-1)),\quad
r\to\infty.$

Therefore, we can write

$\displaystyle f_{R,\mathbf{W}}(r,\mathbf{w})$
$\displaystyle=2^{-d}r^{d-1}\exp(-r)c_{\mathbf{u}_{0}}(\exp(-t\mathbf{z})),$
$\displaystyle\sim
2^{-d}r^{d-1}\mathcal{M}_{\mathbf{u_{0}}}(\exp(-r),\mathbf{w})\exp(-r\lambda_{\mathbf{u}_{0}}(\mathbf{w})),\quad
r\to\infty$
$\displaystyle=2^{-d}(\log(m))^{d-1}\mathcal{M}_{\mathbf{u_{0}}}(1/m,\mathbf{w})m^{-\lambda_{\mathbf{u}_{0}}(\mathbf{w})},$

where $r=\log(m)$. Noting that $dr/dm=m^{-1}$, the integrand for the angular
density is regularly-varying at infinity with index less than $-1$, and hence
the integral is finite by Karamata’s theorem.

In the second case, assume that some component of
$\mathbf{w}=(w_{1},...,w_{d})$ is zero, then the path $\mathbf{u}$ does not
terminate at a corner of the copula. Therefore the copula density is bounded
everywhere along the path and hence $f_{R,\mathbf{W}}(r,\mathbf{w})\leq\beta
r^{d-1}\exp(-r)$ for some $\beta>0$. Hence the angular density is finite in
this case as well.

### Proof of Proposition 5.3

Suppose that $f_{R,\mathbf{W}}(r,\mathbf{w})$ is defined on
$[0,\infty)\times\Sigma$, where $\Sigma\subseteq\mathcal{U}_{1}$. By
definition, for any $r_{1}\in(0,\infty)$ we have
$f_{\mathbf{W}}(\mathbf{w})=\int_{0}^{r_{1}}f_{R,\mathbf{W}}(r,\mathbf{w})dr+\int_{r_{1}}^{\infty}f_{R,\mathbf{W}}(r,\mathbf{w})dr$.
Since $f_{\mathbf{W}}(\mathbf{w})$ is finite and $f_{R,\mathbf{W}}$ is bounded
(no point masses), $\int_{r_{1}}^{\infty}f_{R,\mathbf{W}}(r,\mathbf{w})dr\to
0$ as $r_{1}\to\infty$. Therefore, for all $\mathbf{w}\in\Sigma$ and any
$\varepsilon_{1}>0$ there exists $r_{1}$ such that
$\int_{r_{1}}^{\infty}f_{R,\mathbf{W}}(r,\mathbf{w})dr<\varepsilon_{1}$. Since
$f_{R,\mathbf{W}}$ is continuous and finite for $r\in[0,r_{1}]$, for any
$\mathbf{w}\in\Sigma$ and $\varepsilon_{2}>0$ there exists a $\Delta>0$ and
neighborhood
$N_{\Delta}=\\{\mathbf{z}\in\Sigma:\|\mathbf{w}-\mathbf{z}\|_{1}<\Delta\\}$
such that
$|f_{R,\mathbf{W}}(r,\mathbf{w})-f_{R,\mathbf{W}}(r,\mathbf{z})|<\varepsilon_{2}$
for all $(r,\mathbf{z})\in[0,r_{1}]\times N_{\Delta}$. Now choose
$\varepsilon_{1}$ and $r_{1}$ such that
$\int_{r_{1}}^{\infty}f_{R,\mathbf{W}}(r,\mathbf{z})dr<\varepsilon_{1}$ for
all $\mathbf{z}\in N_{\Delta}$. Then we have

$\displaystyle\left|f_{\mathbf{W}}(\mathbf{w})-f_{\mathbf{W}}(\mathbf{z})\right|$
$\displaystyle=\left|\int_{0}^{r_{1}}f_{R,\mathbf{W}}(r,\mathbf{w})dr-\int_{0}^{r_{1}}f_{R,\mathbf{W}}(r,\mathbf{z})dr+\int_{r_{1}}^{\infty}f_{R,\mathbf{W}}(r,\mathbf{w})dr-\int_{r_{1}}^{\infty}f_{R,\mathbf{W}}(r,\mathbf{z})dr\right|$
$\displaystyle<\left|\int_{0}^{r_{1}}f_{R,\mathbf{W}}(r,\mathbf{w})dr-\int_{0}^{r_{1}}f_{R,\mathbf{W}}(r,\mathbf{z})dr\right|+\varepsilon_{1}$
$\displaystyle<r_{1}\varepsilon_{2}+\varepsilon_{1}$

Note that $r_{1}\to\infty$ as $\varepsilon_{1}\to 0^{+}$. However, for any
given $\varepsilon_{1}$ and $r_{1}$, we can set $\varepsilon_{2}=r_{1}^{-2}$,
so that $r_{1}\varepsilon_{2}\to 0^{+}$ as $\varepsilon_{1}\to 0^{+}$. Hence
$f_{\mathbf{W}}(\mathbf{w})$ is continuous for all $\mathbf{w}\in\Sigma$. If
the upper end point of $R$ is finite, then the continuity follows in the same
way.

### Proof of Lemma 5.4

The angular-radial joint density is given by
$f_{R,\mathbf{W}}(r,\mathbf{w})=2^{-d}r^{d-1}\exp(-r)c(\mathbf{u})$, where
$\mathbf{u}=(F_{L}(rw_{1}),...,F_{L}(rw_{d}))$. For any $r\in[0,\infty)$, we
have $\mathbf{u}\in(0,1)^{d}$. Hence $c(\mathbf{u})$ is continuous and
therefore $f_{R,\mathbf{W}}(r,\mathbf{w})$ is also continuous. The bounds on
$f_{R,\mathbf{W}}(r,\mathbf{w})$ can be established in the same way as in the
proof of Proposition 5.2. For the case that the path $\mathbf{u}$ terminates
at an edge of the copula, the copula density is assumed to be bounded
everywhere along the path. In the case that $\mathbf{u}$ terminates at a
corner of the copula, the assumed asymptotic ARE form (21) multiplied by the
marginal product function $m_{L}(r,\mathbf{w})=2^{-d}\exp(-r)$, ensures that
the angular-radial joint density is bounded as well. Hence
$f_{R,\mathbf{W}}(r,\mathbf{w})$ is bounded everywhere and hence satisfies the
assumptions of Proposition 5.3.

### Proof of Proposition 5.5

From the assumptions of the proposition, the angular density is finite at all
angles. So for $r\to\infty$, the conditional radial density has asymptotic
form

$f_{R|\mathbf{W}}(r|\mathbf{w})\sim\frac{2^{-d}r^{d-1}}{f_{\mathbf{W}}(\mathbf{w})}\mathcal{M}(\exp(-r),\mathbf{w})\exp(-r\lambda(\mathbf{w}))\coloneqq\mathcal{M}^{*}(\exp(-r),\mathbf{w})\exp(-r\lambda(\mathbf{w})).$

Note that $\mathcal{M}^{*}(t,\mathbf{w})$ is also slowly-varying in $t$ at
$0^{+}$. Since $f_{R|\mathbf{W}}(r|\mathbf{w})$ is ultimately monotone, from
the monotone convergence theorem for $r\to\infty$ we have

$\displaystyle\bar{F}_{R|\mathbf{W}}(r|\mathbf{w})$
$\displaystyle=\int_{r}^{\infty}f_{R|\mathbf{W}}(s|\mathbf{w})ds\sim\int_{r}^{\infty}\mathcal{M}^{*}(\exp(-s),\mathbf{w})\exp(-s\lambda(\mathbf{w}))ds=\int_{\exp(r)}^{\infty}\mathcal{M}^{*}(1/t,\mathbf{w})t^{-\lambda(\mathbf{w})-1}dt,$

where we have made the substitution $t=\exp(s)$ in the last step. The
integrand on the RHS is regularly-varying in $t$ with index
$-\lambda(\mathbf{w})-1<-1$ and hence by Karamata’s theorem, the integral is
finite and regularly-varying with index $-\lambda(\mathbf{w})$. Therefore,
there exists a function $\mathcal{L}(t,\mathbf{w})$ that is slowly-varying in
$t$ at $0^{+}$, such that
$\bar{F}_{R|\mathbf{W}}(r|\mathbf{w})=\mathcal{L}(\exp(-r),\mathbf{w})\exp(-r\lambda(\mathbf{w}))$.
So for all $r>0$ we have

$\displaystyle\frac{\bar{F}_{R|\mathbf{W}}(\mu+r|\mathbf{w})}{\bar{F}_{R|\mathbf{W}}(\mu|\mathbf{w})}=\frac{\mathcal{L}(\exp(-(\mu+r)),\mathbf{w})}{\mathcal{L}(\exp(-\mu),\mathbf{w})}\frac{\exp(-(\mu+r)\lambda(\mathbf{w}))}{\exp(-\mu\lambda(\mathbf{w}))}\sim\exp(-r\lambda(\mathbf{w})),\quad\mu\to\infty.$

Therefore, assumptions (A1) and (A2) are satisfied taking $\xi(\mathbf{w})=0$
and $\sigma(\mu,\mathbf{w})=1/\lambda(\mathbf{w})$.

### Proof of Proposition 5.7

To demonstrate that the angular density is finite, we need to consider the
behaviour of the joint density in the lower and upper tails. Taking a similar
approach to the proof of Proposition 5.1, in the lower tail we have

$f_{R,\mathbf{W}}(r,\mathbf{w})\sim
b_{\mathbf{0}_{d}}\left(\mathbf{w}\right)\mathcal{L}_{\mathbf{0}_{d}}\left(r\right)r^{\kappa_{\mathbf{0}_{d}}-1},\quad
r\to 0^{+}.$

Hence $f_{R,\mathbf{W}}(r,\mathbf{w})$ is bounded for $r\to 0^{+}$. Moreover,
since the copula density is bounded away from the corners,
$f_{R,\mathbf{W}}(r,\mathbf{w})$ is bounded for any finite value of $r$. For
the upper tail, we note that when the polar origin is defined at
$\mathbf{x}=\mathbf{1}_{d}$ on Pareto margins and $\xi_{m}=1$,
$\delta_{P}(r,\mathbf{w})=\delta_{GP}(r,\mathbf{w})$. Hence from Proposition
4.4, for $\mathbf{w}\in\mathcal{U}_{1}\cap(0,1]^{d}$ we have,

$\delta_{P}(r,\mathbf{w})\sim
s_{\mathbf{w}}^{\kappa_{\mathbf{1}_{d}}-d}b_{\mathbf{1}_{d}}\left((s_{\mathbf{w}}\mathbf{w})^{-1}\right)\mathcal{L}_{\mathbf{1}_{d}}\left(r^{-1}\right)r^{d-\kappa_{\mathbf{1}_{d}}},\quad
r\to\infty.$

For the marginal product function we have

$m_{P}(r,\mathbf{w})=\prod_{j=1}^{d}(1+rw_{j})^{-2}\sim
r^{-2d}\prod_{j=1}^{d}w_{j}^{-2},\quad r\to\infty.$

Combining this we get

$f_{R,\mathbf{W}}(r,\mathbf{w})\sim\left[\prod_{j=1}^{d}w_{j}^{-2}\right]s_{\mathbf{w}}^{\kappa_{\mathbf{1}_{d}}-d}b_{\mathbf{1}_{d}}\left((s_{\mathbf{w}}\mathbf{w})^{-1}\right)\mathcal{L}_{\mathbf{1}_{d}}\left(r^{-1}\right)r^{-1-\kappa_{\mathbf{1}_{d}}},\quad
r\to\infty.$

So $f_{R,\mathbf{W}}\in RV_{-1-\kappa_{\mathbf{1}_{d}}}(\infty)$. Hence by
Karamata’s theorem, $f_{\mathbf{W}}(\mathbf{w})$ is finite and
$\bar{F}_{R|\mathbf{W}}\in RV_{-\kappa_{\mathbf{1}_{d}}}(\infty)$. Thus from
the regular variation property

$\lim_{\mu\to\infty}\frac{\bar{F}_{R|\mathbf{W}}(\mu+r|\mathbf{w})}{\bar{F}_{R|\mathbf{W}}(\mu|\mathbf{w})}=\left(1+\frac{r}{\mu}\right)^{-\kappa_{\mathbf{1}_{d}}}=\bar{F}_{GP}\left(r,\frac{1}{\kappa_{\mathbf{1}_{d}}},\frac{\mu}{\kappa_{\mathbf{1}_{d}}}\right).$

Therefore, assumptions (A1) and (A2) are satisfied, taking
$\xi(\mathbf{w})=1/\kappa_{\mathbf{1}_{d}}$ and
$\sigma(\mu,\mathbf{w})=\mu/\kappa_{\mathbf{1}_{d}}$.

### Proof of Proposition 5.8

From the discussion at the end of Section 3.2, we know that for two-
dimensional angular-radial densities in $L^{1}$ polar coordinates, we have
$f_{R,\mathbf{W}}(r,(1-q,q)))=f_{R,Q}(r,q)$ for $q\in[0,1]$ and
$r\in[\max(q^{-1},(1-q)^{-1}),\infty)$. Using Proposition 5.7 we have

$f_{R,Q}(r,q)\sim(q(1-q))^{-\kappa_{\mathbf{1}_{d}}}b_{\mathbf{1}_{d}}\left(1-q,q\right)\mathcal{L}_{\mathbf{1}_{d}}\left(r^{-1}\right)r^{-1-\kappa_{\mathbf{1}_{d}}},\quad
r\to\infty.$

From Proposition 4.1 we have
$b_{\mathbf{1}_{d}}(1-q,q)=\gamma(q(1-q))^{\tfrac{\kappa_{\mathbf{1}_{d}}}{2}-1}$
for some $\gamma>0$. Substituting this into the equation above completes the
proof.

##  Appendix B Examples of angular-radial models for copulas

In this appendix we present some examples of the various angular-radial models
for copulas introduced in Section 4. Where applicable, we also confirm the
relations between the various models defined in Propositions 4.1 and 4.2.

###  B.1 Extreme value copulas

Extreme value (EV) copulas can be written in the form

$C(\mathbf{u})=\exp(-A(\mathbf{x})),$

where $\mathbf{x}=(x_{1},...,x_{d})=(-\log(u_{1}),...,-\log(u_{d}))$, and
$A:[0,\infty)^{d}\to[0,\infty)$ is a convex homogeneous function of order one,
known as the stable tail dependence function [8], that satisfies
$\max(x_{1},...x_{d})\leq A(x_{1},...,x_{d})\leq x_{1}+\cdots+x_{d}$. The
lower and upper bounds in the inequality correspond to the cases of complete
dependence and independence respectively. In general we will rule out these
cases and assume that the inequality is strict. The homogeneity property of
$A$ suggests an angular-radial representation. Define
$\tau=\|\mathbf{x}\|_{1}$ and $\bm{\psi}=\mathbf{x}/\mathbf{\tau}$. The
variables $\tau$ and $\bm{\psi}$ are referred to as the Pickands coordinates
[49], and are the $L^{1}$ polar coordinates of $\mathbf{x}$. EV copulas can
then be expressed as

$C(\mathbf{u})=\exp(-\tau A(\bm{\psi})).$

The general expression for the density involves many products of partial
derivatives of $A$. However, the following general form is useful for
asymptotic calculations in the upper and lower tail. Progressively taking
partial derivatives shows that

$\displaystyle\frac{\partial}{\partial u_{1}}C(\mathbf{u})$
$\displaystyle=-\frac{1}{u_{1}}\frac{\partial}{\partial
x_{1}}\exp(-A(\mathbf{x}))=\frac{1}{u_{1}}\exp(-A(\mathbf{x}))A^{(1,0,0,...)}(\mathbf{x})$
$\displaystyle\frac{\partial^{2}}{\partial u_{1}\partial u_{2}}C(\mathbf{u})$
$\displaystyle=-\frac{1}{u_{1}u_{2}}\frac{\partial}{\partial
x_{2}}\left[\exp(-A(\mathbf{x}))A^{(1,0,0,...)}(\mathbf{x})\right]$
$\displaystyle=\frac{1}{u_{1}u_{2}}\exp(-A(\mathbf{x}))\left[A^{(1,0,0,...)}(\mathbf{x})A^{(0,1,0,...)}(\mathbf{x})-A^{(1,1,0,...)}(\mathbf{x})\right]$
$\displaystyle\;\;\vdots$ $\displaystyle\frac{\partial^{d}}{\partial
u_{1}\cdots\partial u_{d}}C(\mathbf{u})$
$\displaystyle=\left(\prod_{j=1}^{d}u_{j}^{-1}\right)\exp(-A(\mathbf{x}))\left[\left(\prod_{j=1}^{d}A_{j}(\mathbf{x})\right)+...+(-1)^{d-1}A^{(1,...,1)}(\mathbf{x})\right],$

where $A_{j}=\partial A/\partial x_{j}$ and the notation
$A^{(n_{1},...,n_{d})}$ denotes partial differentiation $n_{j}$ times in the
$j^{\text{th}}$ variable. The omitted terms on the final line involve products
of partial derivatives of $A$ of mixed order. Note that $n^{\text{th}}$-order
partial derivatives of $A$ are homogeneous of order $1-n$, and that
$\prod_{j=1}^{d}u_{j}^{-1}=\exp(\tau)$. We can therefore write

$c(\mathbf{u})=\exp(-\tau(A(\bm{\psi})-1))\left[\left(\prod_{j=1}^{d}A_{j}(\bm{\psi})\right)+\tau^{-1}[...]+\tau^{-2}[...]+...+(-\tau)^{1-d}A^{(1,...,1)}(\bm{\psi})\right],$
(33)

where the omitted terms involve partial derivatives of $A$ of orders between
$2$ and $d-1$, evaluated at $\bm{\psi}$.

####  B.1.1 ARL model

Lower tail: In the lower tail we have $C(t,...,t)=t^{A(\mathbf{1}_{d})}$.
Hence EV copulas have lower tail order
$\kappa_{\mathbf{0}_{d}}=A(\mathbf{1}_{d})$ with
$\mathcal{L}_{\mathbf{0}_{d}}(t)\equiv 1$. [7] noted that in the lower tail

$\displaystyle C(t\mathbf{w})$
$\displaystyle=\exp\left(\log(t)A\left(1+\frac{\log(w_{1})}{\log(t)},...,1+\frac{\log(w_{d})}{\log(t)}\right)\right)$
$\displaystyle\sim\exp\left(\log(t)\left[A(\mathbf{1}_{d})+A_{1}(\mathbf{1}_{d})\frac{\log(w_{1})}{\log(t)}+\cdots+A_{d}(\mathbf{1}_{d})\frac{\log(w_{d})}{\log(t)}\right]\right),\quad\log(t)\to
0^{+}$
$\displaystyle=t^{A(\mathbf{1}_{d})}\prod_{j=1}^{d}w_{j}^{A_{j}(\mathbf{1}_{d})}.$

Hence EV copulas have lower tail order function
$B_{\mathbf{0}_{d}}(\mathbf{w})=\prod_{j=1}^{d}w_{j}^{A_{j}(\mathbf{1}_{d})}$.
However, note that the asymptotic equivalence above assumes that
$\log(w_{j})/\log(t)\to 0^{+}$ as $t\to 0^{+}$ for $j=1,...,d$, so this form
of the copula does not hold if $w_{j}$ decreases at a similar rate to $t$.
This is discussed further in Section 5.3. Using (19) and differentiating the
expression for $C(t\mathbf{w})$, we obtain for the density

$c(t\mathbf{w})\sim\left[\prod_{j=1}^{d}A_{j}(\mathbf{1}_{d})\right]t^{A(\mathbf{1}_{d})-d}\prod_{j=1}^{d}w_{j}^{A_{j}(\mathbf{1}_{d})-1}.$
(34)

Alternatively, we can work directly with the density, using the same expansion
as above to obtain

$\exp(-\tau(A(\bm{\psi})-1))\sim
t^{A(\mathbf{1}_{d})-d}\prod_{j=1}^{d}w_{j}^{A_{j}(\mathbf{1}_{d})-1}.$

For the terms in the square brackets in (33), we note that in the lower tail

$\displaystyle\tau$
$\displaystyle=-\log\left(t^{d}\prod_{j=1}^{d}w_{j}\right)\to\infty,\quad t\to
0^{+}$ $\displaystyle\bm{\psi}$
$\displaystyle\to\left(\tfrac{1}{d},...,\tfrac{1}{d}\right),\quad t\to 0^{+}.$

Therefore, the terms in the square brackets in (33) which are proportional to
$\tau^{-1}$, $\tau^{-2}$, …, $\tau^{1-d}$, tend to zero. The remaining terms
are all homogeneous of order zero, so
$\prod_{j=1}^{d}A_{j}(\mathbf{1}_{d})=\prod_{j=1}^{d}A_{j}\left(\tfrac{1}{d},...,\tfrac{1}{d}\right)$,
which agrees with the asymptotic expression obtained from the tail order
function.

Upper tail: In the upper tail, for $\mathbf{u}=\mathbf{1}_{d}-t\mathbf{w}$ and
$\mathbf{w}\in\mathcal{U}_{1}\cap[0,1]^{d}$, for $t\to 0^{+}$ we have

$\displaystyle\tau$
$\displaystyle=-\sum_{j=1}^{d}\log\left(1-tw_{j}\right)=\sum_{j=1}^{d}(tw_{j}+O(t^{2}))=t+O(t^{2}),$
$\displaystyle\bm{\psi}$
$\displaystyle=\tau^{-1}(-\log(1-tw_{1}),...,-\log(1-tw_{d}))=\mathbf{w}+O(t).$

The bounds on $A$ together with the convexity property imply that
$A(\mathbf{w}+O(t))=A(\mathbf{w})+O(t)$. Hence

$C(\mathbf{1}_{d}-t\mathbf{w})=\exp(-tA(\mathbf{w})+O(t^{2}))=1-tA(\mathbf{w})+O(t^{2}).$
(35)

The general relation between asymptotic forms of
$C(\mathbf{1}_{d}-t\mathbf{w})$ and $C_{\mathbf{1}_{d}}(t\mathbf{w})$ is
discussed in [29]. For the two-dimensional case, for any copula we have
$C_{(1,1)}(t(1-w),tw)=t-1+C(1-t(1-w),1-tw)$, and hence for EV copulas
$C_{(1,1)}(t\mathbf{w})\sim t(1-A(\mathbf{w}))$ for $t\to 0^{+}$. For the
copula density, since the partial derivatives of $C$ are ultimately monotone,
we can differentiate (35) to obtain

$c_{\mathbf{1}_{d}}(t\mathbf{w})\sim
t^{1-d}\left|A^{(1,...,1)}(\mathbf{w})\right|.$

Hence EV copulas have upper tail order $\kappa_{\mathbf{1}_{d}}=1$, although
the expression above does not give separate values for
$\Upsilon_{\mathbf{1}_{d}}$ and $b_{\mathbf{1}_{d}}(\mathbf{w})$. As with the
lower tail order function, this asymptotic form can be obtained directly from
the density. Since $\tau=t+O(t^{2})$, $\exp(-\tau(A(\bm{\psi})-1))\to 1$ as
$t\to 0^{+}$. Similarly, the dominant terms in the square brackets in (33) are
those proportional to $\tau^{1-d}$, giving the same asymptotic expression
obtained by differentiating (35).

####  B.1.2 ARE model

Lower tail: In the lower tail we have $u_{j}=\exp(-rw_{j})$ and hence for
$\mathbf{w}\in\mathcal{U}_{1}\cap[0,1]^{d}$ we have $\tau=r$ and
$\bm{\psi}=\mathbf{w}$. Therefore

$C(\exp(-r\mathbf{w}))=\exp(-rA(\mathbf{w})),$

and hence $\Lambda_{\mathbf{0}_{d}}(\mathbf{w})=A(\mathbf{w})$, and
$\mathcal{L}_{\mathbf{0}_{d}}(t,\mathbf{w})\equiv 1$. For the density, we note
that the terms in the square brackets of (33) that are proportional to
$\tau^{-1}$, $\tau^{-2}$, etc., all tend to zero for $r\to\infty$. Therefore

$c(\exp(-r\mathbf{w}))\sim\left(\prod_{j=1}^{d}A_{j}(\mathbf{w})\right)\exp(-r(A(\mathbf{w})-1)),\quad
r\to\infty.$ (36)

Hence we have $\lambda_{\mathbf{0}_{d}}(\mathbf{w})=A(\mathbf{w})$, and
$\mathcal{M}_{\mathbf{0}_{d}}(t,\mathbf{w})=\prod_{j=1}^{d}A_{j}(\mathbf{w})$.

The copula and copula density exponent functions satisfy the assumptions of
Proposition 4.1. We can confirm that the expressions for the tail order and
tail density functions derived in the previous subsection are in agreement
with the forms given in Proposition 4.1. As noted above,
$\kappa_{(0,0)}=A(1,1)$. Also note that,
$\beta\coloneqq[\tfrac{d}{dw}\Lambda_{(0,0)}(1-w,w)]_{w=1/2}=A_{2}(\tfrac{1}{2},\tfrac{1}{2})-A_{1}(\tfrac{1}{2},\tfrac{1}{2})=A_{2}(1,1)-A_{1}(1,1)$,
where the final equality is obtained by using the fact that since $A$ is
homogeneous of order one, its first partial derivatives are homogeneous of
order zero. Next note that by Euler’s theorem on homogeneous functions
$A(1,1)=A_{1}(1,1)+A_{2}(1,1)$. Substituting these expressions into the form
of the tail order and tail density functions given in Proposition 4.1, shows
that this agrees with the expressions derived above.

Upper tail: Since EV copulas have an upper tail order of 1,
$\Lambda_{\mathbf{1}_{d}}(\mathbf{w})=\max(w_{1},...,w_{d})$. The case for the
density exponent function is more complicated. Examples of
$\delta_{L}(r,\mathbf{w})$ (which is related to
$\lambda_{\mathbf{1}_{d}}(\mathbf{w})$ by Proposition 4.3) for two-dimensional
cases are considered in the following subsection.

####  B.1.3 AR copula function for Laplace margins

For the two-dimensional case we write

$c(u,v)=T_{1}(T_{2}+T_{3}),$ (37)

where

$\displaystyle T_{1}$ $\displaystyle=\exp(-\tau(A(\bm{\psi})-1)),$
$\displaystyle T_{2}$
$\displaystyle=A^{(1,0)}(\bm{\psi})A^{(0,1)}(\bm{\psi}),$ $\displaystyle
T_{3}$ $\displaystyle=-\tau^{-1}A^{(1,1)}(\bm{\psi}).$

To understand the asymptotic behave of the copula density, we consider the
asymptotic behaviour of the Pickands coordinates. On Laplace margins with
$(u,v)=(F_{L}(rw_{1}),F_{L}(rw_{2}))$,
$\mathbf{w}=(w_{1},w_{2})=(\cos_{1}(q),\sin_{1}(q))$, and
$\bm{\psi}=(\psi,1-\psi)$, we have

$\tau=\begin{cases}\log(4)+r&q\in[-2,-1]\\\
\log(2)+r|w_{2}|+O(\exp(-r|w_{1}|)&q\in(-1,0]\\\
\tfrac{1}{2}\exp(-r|w_{2}|)+O(\exp(-r\min(2|w_{2}|,|w_{1}|))&q\in(0,1/2)\\\
\exp(-r/2)+O(\exp(-r))&q=1/2\\\
\tfrac{1}{2}\exp(-r|w_{1}|)+O(\exp(-r\min(2|w_{1}|,|w_{2}|))&q\in(1/2,1)\\\
\log(2)+r|w_{1}|+O(\exp(-r|w_{2}|)&q\in[1,2)\end{cases}$
$\psi=\begin{cases}|w_{2}|+O(r^{-1})&q\in[-2,-1]\\\
1-O((r|w_{2}|)^{-1}\exp(-r|w_{1}|))&q\in(-1,0)\\\
1-O(\exp(-r(|w_{1}|-|w_{2}|))&q\in[0,1/2)\\\ 1/2&q=1/2\\\
O(\exp(-r(|w_{2}|-|w_{1}|))&q\in(1/2,1]\\\
O((r|w_{1}|)^{-1}\exp(-r|w_{2}|))&q\in(1,2)\end{cases}$

The variation of $(\tau,\psi)$ with $(r,q)$ is illustrated in Figure 16. From
the preceding subsection and Proposition 4.3, we know the asymptotic behaviour
of $\delta_{L}(r,\mathbf{w})$ in the third quadrant. In the second and fourth
quadrants, $\psi$ is exponentially close to zero or one, so since
$A(0,1)=A(1,0)=1$ for all stable tail dependence functions, $T_{1}\to 1$ as
$r\to\infty$. Similarly, in the first quadrant $\tau\to 0$, and hence
$T_{1}\to 1$ as $r\to\infty$ in this quadrant as well. Therefore, in the
first, second and fourth quadrants, the asymptotic behaviour of
$\delta_{L}(r,\mathbf{w})$ is determined by the terms $T_{2}$ and $T_{3}$. To
illustrate various possibilities for how $T_{2}$ and $T_{3}$ behave in these
quadrants, and hence the asymptotic behaviour of $\delta_{L}(r,\mathbf{w})$,
we consider three examples for the stable tail dependence function $A$: the
symmetric and asymmetric logistic models, and the Hüsler-Reiss model.

Figure 16: Pickands coordinates $(t,\psi)$ for two-dimensional EV copulas as a
function of $L^{1}$ polar coordinates $(r,q)$ on Laplace margins.

Symmetric logistic: The symmetric logistic model with parameter $\alpha\geq 1$
is given by

$A(\bm{\psi})=z^{\tfrac{1}{\alpha}},$

with $z=\psi^{\alpha}+(1-\psi)^{\alpha}$. The case $\alpha=1$ corresponds to
the independence copula. When $\alpha>1$, we have

$\displaystyle T_{2}$
$\displaystyle=z^{\tfrac{2}{\alpha}-2}(\psi(1-\psi))^{\alpha-1}$
$\displaystyle T_{3}$
$\displaystyle=(\alpha-1)z^{-\tfrac{1}{\alpha}}\tau^{-1}T_{2}.$

After substituting the expressions for $\tau$ and $\psi$ above we find that
$\delta_{L}(r,\mathbf{w})$ has asymptotic form (25) with

$\displaystyle\beta(\mathbf{w})$
$\displaystyle=\begin{cases}0,&q\in(-2,-1)\cup[0,1],\\\
1-\alpha,&q\in[-1,0)\cup(1,2],\end{cases}$ $\displaystyle\lambda(\mathbf{w})$
$\displaystyle=\begin{cases}A(\mathbf{w}),&q\in(-2,-1],\\\
1+(\alpha-1)|w_{1}|,&q\in(-1,0],\\\
1-\alpha+(2\alpha-1)\max(|w_{1}|,|w_{2}|),&q\in(0,1]\\\
1+(\alpha-1)|w_{2}|&q\in(1,2].\end{cases}$

Note that in this case
$A^{(1,1)}(\psi,1-\psi)=(1-\alpha)z^{\tfrac{1}{\alpha}-2}(\psi(1-\psi))^{\alpha-1}$,
so as $\psi\to 0^{+}$ we have
$A^{(1,1)}(\psi,1-\psi)\sim(1-\alpha)\psi^{\alpha-1}$. Therefore, the
assumptions of Proposition 4.2 are satisfied, taking
$\beta_{1}=\beta_{2}=\alpha-1$. It can then be verified that Proposition 4.2
in combination with Proposition 4.3 gives the same form of
$\lambda(\mathbf{w})$ obtained above for the range $q\in(0,1)$.

Asymmetric logistic: The asymmetric logistic model proposed by [42], is given
by

$A(\bm{\psi})=(1-\gamma_{1})\psi+(1-\gamma_{2})(1-\psi)+z^{\tfrac{1}{\alpha}},$

where $z=(\gamma_{1}\psi)^{\alpha}+(\gamma_{2}(1-\psi))^{\alpha}$ and
$\alpha\geq 1$ and $\gamma_{1},\gamma_{2}\in[0,1]$. When
$\gamma_{1}=\gamma_{2}=1$ this reduces to the symmetric logistic model.
Independence occurs when $\alpha=1$ or $\gamma_{1}=0$ or $\gamma_{2}=0$. In
the case that $\gamma_{1},\gamma_{2}\in(0,1)$, we have

$\displaystyle T_{2}$
$\displaystyle=\left(1-\gamma_{1}+\gamma_{1}^{\alpha}\psi^{\alpha-1}z^{\tfrac{1}{\alpha}-1}\right)\left(1-\gamma_{2}+\gamma_{2}^{\alpha}(1-\psi)^{\alpha-1}z^{\tfrac{1}{\alpha}-1}\right)$
$\displaystyle T_{3}$
$\displaystyle=(\alpha-1)(\gamma_{1}\gamma_{2})^{\alpha}(\psi(1-\psi))^{\alpha-1}\tau^{-1}z^{\tfrac{1}{\alpha}-2}.$

For $q\in[-1,1/2)$, $\psi\to 1$ as $r\to\infty$ and hence $T_{2}\to
1-\gamma_{2}$ as $r\to\infty$. Similarly, for $q\in(1/2,2]$, $T_{2}\to
1-\gamma_{1}$ as $r\to\infty$. For $q$ in the second and fourth quadrants,
$\tau\to\infty$ as $r\to\infty$, hence $T_{3}\to 0$. In the first quadrant for
$r\to\infty$ we have

$T_{3}\sim
2(\alpha-1)\exp(r[1-\alpha+(2\alpha-1)\min(w_{1},w_{2})])\times\begin{cases}\gamma_{1}^{\alpha}\gamma_{2}^{1-\alpha},&q<1/2,\\\
\gamma_{2}^{\alpha}\gamma_{1}^{1-\alpha},&q>1/2.\end{cases}$

The terms in the exponential function are positive when
$\min(w_{1},w_{2})>(\alpha-1)/(2\alpha-1)$. Overall, we find that
$\delta_{L}(r,\mathbf{w})$ has asymptotic form (25) with $\beta(\mathbf{w})=0$
for all $q$ and

$\displaystyle\lambda(\mathbf{w})$
$\displaystyle=\begin{cases}A(\mathbf{w}),&q\in(-2,-1),\\\
1&q\in\left[-1,\dfrac{\alpha-1}{2\alpha-1}\right],\\\\[10.0pt]
1-\alpha+(2\alpha-1)\max(|w_{1}|,|w_{2}|),&q\in\left(\dfrac{\alpha-1}{2\alpha-1},\dfrac{\alpha}{2\alpha-1}\right)\\\\[10.0pt]
1&q\in\left[\dfrac{\alpha}{2\alpha-1},2\right].\end{cases}$

Note that although $A^{(1,1)}$ has the same form as for the symmetric logistic
model, the assumptions of Proposition 4.2 are not satisfied, since $T_{3}\to
0^{+}$ for $q\to 0^{+}$ and $q\to 1^{-}$, but $T_{2}\to 1-\gamma_{1}$ or
$1-\gamma_{2}$, respectively. Hence the ARL model is not valid for $q\to
0^{+}$ or $q\to 1^{-}$.

In the first quadrant $\lambda(\mathbf{w})$ is symmetric about $q=1/2$ and is
not dependent on the values of $\gamma_{1}$ and $\gamma_{2}$. However, at
finite levels, the corresponding isodensity contours are not symmetric about
$q=1/2$, as illustrated in Figure 12, for a case with $\alpha=5$,
$\gamma_{1}=0.9$ and $\gamma_{2}=0.1$. To improve the agreement between the
SPAR approximation and true joint density at finite levels, we can define the
angular-radial coordinate system relative to a different origin. Note that a
finite shift in the origin does not affect the asymptotic behaviour of
$\delta_{L}(r,\mathbf{w})$, and hence the values of $\beta(\mathbf{w})$ and
$\lambda(\mathbf{w})$ remain unchanged.

Define new angular-radial variables relative to origin $(x_{0},y_{0})$,
$r^{*}=(x-x_{0})+(y-y_{0})$ and $q^{*}=(y-y_{0})/r^{*}$, for $x>x_{0}$ and
$y>y_{0}$. Then, in terms of the original angular-radial coordinates,

$\displaystyle r$ $\displaystyle=r^{*}+x_{0}+y_{0},$ $\displaystyle q$
$\displaystyle=\frac{r^{*}q^{*}+y_{0}}{r}.$

We can find $(x_{0},y_{0})$ such that $T_{2}\sim T_{3}$ as $r\to\infty$ at
constant angle $q^{*}$. This gives two sets of equations

$\displaystyle 1-\gamma_{1}$
$\displaystyle=2(\alpha-1)\gamma_{1}^{\alpha}\gamma_{2}^{1-\alpha}\exp(r[1-\alpha+(2\alpha-1)q]),\quad
q\in(0,1/2)$ $\displaystyle 1-\gamma_{2}$
$\displaystyle=2(\alpha-1)\gamma_{2}^{\alpha}\gamma_{1}^{1-\alpha}\exp(r[\alpha+(1-2\alpha)q]),\quad
q\in(1/2,1).$

Substituting the new coordinates and rearranging gives

$\displaystyle\beta_{1}$
$\displaystyle=r^{*}(1-\alpha+(2\alpha-1)q^{*})+(1-\alpha)x_{0}+\alpha
y_{0},\quad q\in(0,1/2)$ $\displaystyle\beta_{2}$
$\displaystyle=r^{*}(\alpha+(1-2\alpha)q^{*})+\alpha
x_{0}+(1-\alpha)y_{0},\quad q\in(1/2,1),$

where

$\displaystyle\beta_{1}$
$\displaystyle=\log\left(\frac{1-\gamma_{1}}{2(\alpha-1)\gamma_{2}}\left(\frac{\gamma_{2}}{\gamma_{1}}\right)^{\alpha}\right),$
$\displaystyle\beta_{2}$
$\displaystyle=\log\left(\frac{1-\gamma_{2}}{2(\alpha-1)\gamma_{1}}\left(\frac{\gamma_{1}}{\gamma_{2}}\right)^{\alpha}\right).$

For these equations to hold approximately for $r^{*}\to\infty$, the terms
multiplying $r^{*}$ must be zero. Then solving for $(x_{0},y_{0})$ gives

$\displaystyle
x_{0}=\frac{(\alpha-1)\beta_{1}+\alpha\beta_{2}}{2\alpha-1},\quad
y_{0}=\frac{\alpha\beta_{1}+(\alpha-1)\beta_{2}}{2\alpha-1}.$

The effect of the choice of origin on the speed of convergence of
$\delta_{L}(r,\mathbf{w})$ to its asymptotic form is illustrated in Figure 17
for the case $\alpha=5$, $\gamma_{1}=0.9$ and $\gamma_{2}=0.1$. The plots show
that placing the origin at $(0,0)$ results in slow convergence, whereas
placing the origin so that $T_{2}\sim T_{3}$ at constant angle in the first
quadrant improves the rate of convergence.

Figure 17: Effect of choice of origin for polar coordinates on speed of
convergence of angular-radial dependence function to asymptotic form, for EV
copula with asymmetric logistic dependence with $\alpha=0.5$, $\gamma_{1}=0.9$
and $\gamma_{2}=0.1$. Left plot has polar origin at $(x_{0},y_{0})=(0,0)$,
right plot has origin selected such that $T_{2}=T_{3}$ at fixed angle – see
(37).

Hüsler-Reiss: The Hüsler-Reiss dependence model [35], with parameter
$\alpha>0$, is given by

$A(\bm{\psi})=\psi z_{1}+(1-\psi)z_{2},$

where

$\displaystyle z_{1}$
$\displaystyle=\Phi\left(\frac{1}{\alpha}+\frac{\alpha}{2}\log\left(\frac{\psi}{1-\psi}\right)\right),$
$\displaystyle z_{2}$
$\displaystyle=\Phi\left(\frac{1}{\alpha}-\frac{\alpha}{2}\log\left(\frac{\psi}{1-\psi}\right)\right),$

and $\Phi$ is the standard normal distribution function. In this case we have

$\displaystyle T_{2}$ $\displaystyle=z_{1}z_{2},$ $\displaystyle T_{3}$
$\displaystyle=\frac{\alpha}{2\tau\psi}\,\phi\left(\frac{1}{\alpha}-\frac{\alpha}{2}\log\left(\frac{\psi}{1-\psi}\right)\right),$

where $\phi$ is the standard normal density function. In this case we have

$\displaystyle\left|A^{(1,1)}(\psi,1-\psi)\right|$
$\displaystyle=\frac{\alpha}{2\psi}\,\phi\left(\frac{1}{\alpha}-\frac{\alpha}{2}\log\left(\frac{\psi}{1-\psi}\right)\right)$
$\displaystyle=\phi\left(\frac{1}{\alpha}\right)\frac{\alpha}{2\psi}\,\exp\left(\log\left(\frac{\psi}{1-\psi}\right)-\frac{\alpha^{2}}{8}\log^{2}\left(\frac{\psi}{1-\psi}\right)\right)$
$\displaystyle=\phi\left(\frac{1}{\alpha}\right)\frac{\alpha}{2\psi}\left(\frac{\psi}{1-\psi}\right)^{1-\tfrac{\alpha^{2}}{8}\log\left(\tfrac{\psi}{1-\psi}\right)}.$

This is rapidly-varying for $\psi\to 0^{+}$ and $\psi\to 1^{-}$ with index
$+\infty$. The assumptions of Proposition 4.2 are therefore not satisfied, so
we cannot obtain the copula density exponent function from the tail density
function. Instead, we can calculate the AR copula function directly.

To calculate asymptotic expressions for $T_{2}$ we utilise the asymptotic
Mill’s ratio for the standard normal distribution. Combining this with the
other terms, we obtain the following results

$\delta_{L}(r,\mathbf{w})\sim\begin{cases}A_{1}\exp\left(-\dfrac{1}{2}\left[\left(\dfrac{\alpha}{2}r\right)^{2}+\left(\dfrac{\alpha^{2}}{2}\log(\log(4))-1\right)r\right]\right),&q=0,1\\\
A_{2}\exp\left(-\dfrac{1}{2}\left[\left(\dfrac{\alpha}{2}(w_{1}-w_{2})r\right)^{2}-r\right]\right),&q\in(0,1)\\\
A_{3}\sqrt{\dfrac{2}{r}}\exp\left(-\dfrac{1}{2}\left[\left(\dfrac{\alpha}{2}r|w_{2}|\right)^{2}+\left(\dfrac{\alpha^{2}}{2}\log(2r|w_{1}|)-1\right)r|w_{2}|+\left(\dfrac{\alpha}{2}\log(2r|w_{1}|)\right)^{2}\right]\right),&q\in(1,2)\\\
A_{4}\,(\log(r))^{-1}\,r^{\tfrac{1}{4}\alpha^{2}\log(\log(2))+\tfrac{1}{2}}\,\exp\left(-\dfrac{1}{2}\left(\dfrac{1}{2}\alpha\log(r)\right)^{2}\right),&q=2,3\\\
A_{5}\exp(-r(A(\mathbf{w})-1)),&q\in(2,3),\\\ \end{cases}$

where the terms $A_{j}$ are constants dependent on $q$ only, given by

$\displaystyle A_{1}$
$\displaystyle=\alpha\phi\left(\frac{1}{\alpha}\right)(\log(4))^{-\tfrac{1}{2}-\tfrac{\alpha^{2}}{8}\log(\log(4))}$
$\displaystyle A_{2}$ $\displaystyle=\alpha\phi\left(\frac{1}{\alpha}\right)$
$\displaystyle A_{3}$
$\displaystyle=\phi\left(\frac{1}{\alpha}\right)2^{-\tfrac{\alpha^{2}}{4}\tfrac{|w_{1}|}{|w_{2}|}}\left[\frac{2}{\alpha}\frac{\sqrt{|w_{1}|}}{|w_{2}|}+\frac{\alpha}{2}\frac{1}{\sqrt{|w_{1}|}}\right]$
$\displaystyle A_{4}$
$\displaystyle=\frac{4}{\alpha}\phi\left(\frac{1}{\alpha}\right)(\log(2))^{-\tfrac{1}{2}-\tfrac{\alpha^{2}}{8}\log(\log(2))}$
$\displaystyle A_{5}$
$\displaystyle=4^{1-A(\mathbf{w})}\Phi\left(\frac{1}{\alpha}+\frac{\alpha}{2}\log\left(\frac{|w_{1}|}{|w_{2}|}\right)\right)\Phi\left(\frac{1}{\alpha}-\frac{\alpha}{2}\log\left(\frac{|w_{1}|}{|w_{2}|}\right)\right).$

Clearly, this does not have asymptotic form (25) for all $q$. Moreover,
outside the third quadrant, the limit set is the line segment
$\\{(x,x),x\in[0,1]\\}$. However, after multiplying by
$m_{L}(r,\mathbf{w})=\tfrac{1}{4}\exp(-r)$, the terms multiplying the leading
order contributions within the exponential are continuous in $q$, and hence
assumptions (A1) and (A2) are satisfied.

###  B.2 Gaussian copula

For the Gaussian copula we focus on models for the copula density, since the
copula does not admit a simple closed form expression. The bivariate Gaussian
copula density with Pearson correlation coefficient $\rho\in(-1,1)$ is given
by

$\displaystyle c(u,v)$ $\displaystyle=\frac{1}{\sqrt{1-\rho^{2}}}T_{1}T_{2},$
$\displaystyle T_{1}$
$\displaystyle=\exp(-\tfrac{1}{2}\rho\beta(x^{2}+y^{2})),$ $\displaystyle
T_{2}$ $\displaystyle=\exp(\beta xy),$

where $x=\Phi^{-1}(u)$, $y=\Phi^{-1}(v)$, where $\Phi$ is the standard normal
cdf $\beta=\rho/(1-\rho^{2})$. From the symmetry of the copula, we need only
consider the asymptotic behaviour of the copula in the lower left corner. For
$u\to 0^{+}$, $x\to-\infty$, and utilising the asymptotic Mill’s ratio we have
$\Phi(x)\sim-\phi(x)/x$, where $\phi$ is the standard normal pdf. Solving for
$x$ gives

$\displaystyle x^{2}\sim W_{0}\left(\frac{1}{2\pi(1-u)^{2}}\right)\sim
L_{u}-\log(L_{u}),\quad u\to 0^{+},$

where $W_{0}$ is the zeroth branch of the Lambert W function, and
$L_{u}=-\log(2\pi u^{2})$. Substituting this approximation shows that

$\displaystyle
T_{1}\sim\left(2\pi(1-u)(1-v)\sqrt{L_{u}L_{v}}\right)^{\rho\beta},\quad u,v\to
0^{+}.$ (38)

For the product terms, for $u,v\to 0^{+}$ we have

$\displaystyle xy$
$\displaystyle\sim\sqrt{(L_{u}-\log(L_{u}))(L_{v}-\log(L_{v}))}$
$\displaystyle=\sqrt{L_{u}L_{v}}\sqrt{1-\frac{\log(L_{u})}{L_{u}}-\frac{\log(L_{v})}{L_{v}}+\frac{\log(L_{u})\log(L_{v})}{L_{u}L_{v}}}$
$\displaystyle\sim\sqrt{L_{u}L_{v}}\left[1-\frac{\log(L_{u})}{2L_{u}}-\frac{\log(L_{v})}{2L_{v}}+\frac{\log(L_{u})\log(L_{v})}{2L_{u}L_{v}}\right]$
$\displaystyle\sim\sqrt{L_{u}L_{v}}-\frac{1}{2}\left[\sqrt{\frac{L_{v}}{L_{u}}}\log(L_{u})+\sqrt{\frac{L_{u}}{L_{v}}}\log(L_{v})\right].$
(39)

These expressions can be used to derive the tail density function and copula
density exponent functions, as discussed below.

####  B.2.1 ARL model

In the lower left corner we have $(u,v)=(t(1-w),tw)$, so that

$\displaystyle
L_{u}L_{v}=-4\left(\log^{2}(\sqrt{2\pi}t)+\log(\sqrt{2\pi}t)\log(w(1-w))+\log(w)\log(1-w)\right)$

The terms which dominate depend on the relative sizes of $t$ and $w$. For
fixed $w\in(0,1)$ and $t\to 0^{+}$, we have

$\displaystyle\sqrt{L_{u}L_{v}}$
$\displaystyle=-2\log(\sqrt{2\pi}t)\left(1+\frac{\log(w(1-w))}{\log(\sqrt{2\pi}t)}+\frac{\log(w)\log(1-w)}{\log^{2}(\sqrt{2\pi}t)}\right)^{1/2}$
$\displaystyle\sim-2\log(\sqrt{2\pi}t)\left(1+\frac{\log(w(1-w))}{2\log(\sqrt{2\pi}t)}+\frac{\log(w)\log(1-w)}{2\log^{2}(\sqrt{2\pi}t)}\right)$
$\displaystyle\sim-\log(2\pi t^{2}w(1-w)).$