# On convergence and mass distributions of multivariate Archimedean copulas
and their interplay with the Williamson transform

Thimo M. Kasper, Nicolas Dietrich and Wolfgang Trutschnig  Thimo M. Kasper
University of Salzburg
Department for Artificial Intelligence and Human Interfaces
Hellbrunnerstrasse 34, 5020 Salzburg<EMAIL_ADDRESS>Nicolas Dietrich
University of Salzburg
Department for Artificial Intelligence and Human Interfaces
Hellbrunnerstrasse 34, 5020 Salzburg<EMAIL_ADDRESS>Wolfgang Trutschnig
University of Salzburg
Department for Artificial Intelligence and Human Interfaces
Hellbrunnerstrasse 34, 5020 Salzburg<EMAIL_ADDRESS>
###### Abstract.

Motivated by a recently established result saying that within the class of
bivariate Archimedean copulas standard pointwise convergence implies weak
convergence of almost all conditional distributions this contribution studies
the class $\mathcal{C}_{ar}^{d}$ of all $d$-dimensional Archimedean copulas
with $d\geq 3$ and proves the afore-mentioned implication with respect to
conditioning on the first $d-1$ coordinates. Several properties equivalent to
pointwise convergence in $\mathcal{C}_{ar}^{d}$ are established and - as by-
product of working with conditional distributions (Markov kernels) -
alternative simple proofs for the well-known formulas for the level set masses
$\mu_{C}(L_{t})$ and the Kendall distribution function $F_{K}^{d}$ as well as
a novel geometrical interpretation of the latter are provided. Viewing
normalized generators $\psi$ of $d$-dimensional Archimedean copulas from the
perspective of their so-called Williamson measures $\gamma$ on $(0,\infty)$ is
then shown to allow not only to derive surprisingly simple expressions for
$\mu_{C}(L_{t})$ and $F_{K}^{d}$ in terms of $\gamma$ and to characterize
pointwise convergence in $\mathcal{C}_{ar}^{d}$ by weak convergence of the
Williamson measures but also to prove that regularity/singularity properties
of $\gamma$ directly carry over to the corresponding copula
$C_{\gamma}\in\mathcal{C}_{ar}^{d}$. These results are finally used to prove
the fact that the family of all absolutely continuous and the family of all
singular $d$-dimensional copulas is dense in $\mathcal{C}_{ar}^{d}$ and to
underline that despite of their simple algebraic structure Archimedean copulas
may exhibit surprisingly singular behavior in the sense of irregularity of
their conditional distribution functions.

###### Contents

  1. 1 Introduction
  2. 2 Notation and preliminaries
  3. 3 Markov kernel, mass distribution and Kendall distribution function of multivariate Archimedean copulas
  4. 4 Characterizing pointwise convergence in $\mathcal{C}^{d}_{ar}$ and the interrelation with weak conditional convergence
  5. 5 Archimedean copulas and the Williamson transform
  6. 6 Singular Archimedean copulas with full support
  7. A Level set mass and Kendall distribution function: Calculations
  8. B Approximations by discrete, absolutely continuous and singular Williamson measures

##  1\. Introduction

Archimedean copulas are a well-known family of dependence models whose
popularity is mainly due to their simple algebraic structure: given a so-
called (Archimedean, sufficiently monotone) generator
$\psi:[0,\infty)\to[0,1]=:\mathbb{I}$ and letting $\varphi$ denote its pseudo-
inverse the Archimedean copula $C_{\psi}$ is defined by

$C_{\psi}(x_{1},\ldots,x_{d})=\psi(\varphi(x_{1})+\cdots+\varphi(x_{d})).$

As a consequence, analytic, dependence, and convergence properties of
Archimedean copulas can be characterized in terms of the corresponding
generators (see, e.g., [2, 4, 14, 21, 22]). In particular, it was recently
shown in [14] that within the class of _bivariate_ Archimedean copulas
pointwise convergence is equivalent to uniform convergence of the
corresponding generators and, more importantly, even implies weak convergence
of almost all conditional distributions (a.k.a weak conditional convergence, a
concept generally much stronger than pointwise convergence). This result is
surprising insofar that given samples $(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots$
from some bivariate copula $C$ the corresponding sequence of empirical copulas
$(\hat{E}_{n})_{n\in\mathbb{N}}$ does not necessarily converge weakly
conditional to $C$.
The focus of the current paper is twofold: _on the one hand_ we study
convergence in the class $\mathcal{C}_{ar}^{d}$ of all $d$-dimensional
Archimedean copulas, $d\geq 3$, and show that most results from the bivariate
setting as established in [14] also hold in $\mathcal{C}_{ar}^{d}$, including
the surprising fact that pointwise convergence implies weak conditional
convergence (whereby we consider conditioning on the first $d-1$ coordinates).
As a nice by-product of working with Markov kernels (conditional
distributions) we obtain simple, alternative proofs of the well-known formulas
for the Kendall distribution function $F_{K}^{d}$ and the level set masses of
Archimedean copulas. To the best of the authors’ knowledge, working with so-
called $\ell_{1}$-norm symmetric distributions (as studied in [21]) nowadays
seems to be the standard approach for deriving these formulas in the
multivariate setting - we show that working with Markov kernels constitutes an
interesting alternative and may provide additional insight. Additionally,
motivated by [22, Chapter 4.3] and [4, Section 3] we offer a seemingly novel
geometric interpretation of the level set masses in terms of $\psi$.
And, _on the other hand_ , we revisit the close interrelation between
Archimedean copulas and probability measures $\gamma$ on $(0,\infty)$ via the
so-called Williamson transform as studied in [21] (also see [25]),
characterize properties of the generator $\psi$ in terms of normalized
$\gamma$ (to which we will refer to as Williamson measure) and then prove the
fact that pointwise convergence in $\mathcal{C}_{ar}^{d}$ is equivalent to
weak convergence of the corresponding probability measures on $(0,\infty)$.
Moreover, we derive surprisingly simple expressions for the level set masses
and the Kendall distribution functions in terms of $\gamma$ and then show that
singularity/regularity properties of $\gamma$ directly carry over to the
corresponding Archimedean copula $C_{\gamma}\in\mathcal{C}_{ar}^{d}$. This
very property is then used, firstly, to show that the family of absolutely
continuous as well as two disjoint subclasses of the family of all singular
Archimedean copulas are dense in $\mathcal{C}_{ar}^{d}$ and, secondly, to
illustrate the fact that despite their simple and handy algebraic structure
Archimedean copulas may exhibit surprisingly irregular behavior by
constructing elements of $\mathcal{C}_{ar}^{d}$ which have full support
although being singular (see [4] for the already established bivariate
results).

The rest of this contribution is organized as follows: Section 2 contains
notation and preliminaries that are used throughout the text. Section 3 starts
with deriving an explicit expression for the Markov kernel of $d$-dimensional
Archimedean copulas, then restates the well-known formulas for the masses of
level sets as well as the Kendall distribution function, and provides a
geometric interpretation for the latter in terms of the generator $\psi$. In
Section 4 we derive various characterizations of pointwise convergence in
$\mathcal{C}_{ar}^{d}$ in several steps and prove the main result saying that
pointwise convergence implies weak conditional convergence (with respect to
the first $d-1$ coordinates). Turning to the Williamson transform, in Section
5 we first establish some complementing results on the interrelation of the
generator $\psi$ and the Williamson measure $\gamma$, then characterize
pointwise convergence in $\mathcal{C}_{ar}^{d}$ in terms of weak convergence
of the Williamson measures, and finally show how regularity/singularity
properties of $\gamma$ carry over to $C_{\gamma}$. These properties are then
used to prove the afore-mentioned denseness results and to construct singular
Archimedean copulas with full support. Several examples and graphics
illustrate the obtained results.

##  2\. Notation and preliminaries

In the sequel we will let $\mathcal{C}^{d}$ denote the family of all
$d$-dimensional copulas for some fixed integer $d\geq 3$ and write vectors in
bold symbols. For each copula $C\in\mathcal{C}^{d}$ the corresponding
$d$-stochastic measure will be denoted by $\mu_{C}$, i.e.,
$\mu_{C}([\mathbf{0},\mathbf{x}])=C(\mathbf{x})$ for all
$\mathbf{x}\in\mathbb{I}^{d}$, whereby
$[\mathbf{0},\mathbf{x}]:=[0,x_{1}]\times[0,x_{2}]\times\ldots\times[0,x_{d}]$
and $\mathbb{I}:=[0,1]$. To notation as simple as possible we will frequently
write $\mathbf{x}_{1:m}=(x_{1},\ldots,x_{m})$ for
$\mathbf{x}\in\mathbb{I}^{d}$ and $m\leq d$. Considering $1\leq i<j\leq d$,
the $i$-$j$-marginal of $C$ will be denoted by $C^{ij}$, i.e., we have
$C^{ij}(x_{i},x_{j})=C(1,\ldots,1,x_{i},1,\ldots,1,x_{j},1,\ldots,1)$. In
order to keep notation as simple as possible for every $m<d$ the marginal
copula of the first $m$ coordinates will be denoted by $C^{1:m}$, i.e.,
$C^{1:m}(x_{1},x_{2},\ldots,x_{m})=C(x_{1},x_{2},\ldots,x_{m},1,\ldots,1)$.
Considering the uniform metric $d_{\infty}$ on $\mathcal{C}^{d}$ it is well-
known that $(\mathcal{C}^{d},d_{\infty})$ is a compact metric space and that
in $\mathcal{C}^{d}$ pointwise and uniform convergence are equivalent. For
more background on copulas and $d$-stochastic measures we refer to [3, 22].

For every metric space $(S,d)$ the Borel $\sigma$-field on $S$ will be denoted
by $\mathcal{B}(S)$ and $\mathcal{P}(S)$ denotes the family of all probability
measures on $\mathcal{B}(S)$. The Lebesgue measure on
$\mathcal{B}(\mathbb{I}^{d})$ will be denoted by $\lambda$ or (whenever
particular emphasis to the dimension $d$ is required) by $\lambda_{d}$.
Furthermore $\delta_{x}$ denotes the Dirac measure in $x\in S$.

In what follows Markov kernels will play a prominent role. For $m<d$ an
$m$-_Markov kernel_ from $\mathbb{R}^{m}$ to $\mathbb{R}^{d-m}$ is a mapping
$K:\mathbb{R}^{m}\times\mathcal{B}(\mathbb{R}^{d-m})\rightarrow\mathbb{I}$
fulfilling that for every fixed $E\in\mathcal{B}(\mathbb{R}^{d-m})$ the
mapping $\mathbf{x}\mapsto K(\mathbf{x},E)$ is
$\mathcal{B}(\mathbb{R}^{m})$-$\mathcal{B}(\mathbb{R}^{d-m})$-measurable and
for every fixed $\mathbf{x}\in\mathbb{R}^{m}$ the mapping $E\mapsto
K(\mathbf{x},E)$ is a probability measure on $\mathcal{B}(\mathbb{R}^{d-m})$.
Given a $(d-m)$-dimensional random vector $\mathbf{Y}$ and an $m$-dimensional
random vector $\mathbf{X}$ on a probability space
$(\Omega,\mathcal{A},\mathbb{P})$ we say that a Markov kernel $K(\cdot,\cdot)$
is a regular conditional distribution of $\mathbf{Y}$ given $\mathbf{X}$ if
for every fixed $E\in\mathcal{B}(\mathbb{R}^{d-m})$ the identity

$K(\mathbf{X}(\omega),E)=\mathbb{E}(\mathbf{1}_{E}\circ\mathbf{Y}|\mathbf{X})(\omega)$

holds for $\mathbb{P}$-almost every $\omega\in\Omega$. It is well-known that
for each pair of random vectors $(\mathbf{X},\mathbf{Y})$ as above, a regular
conditional distribution $K(\cdot,\cdot)$ of $\mathbf{Y}$ given $\mathbf{X}$
exists and is unique for $\mathbb{P}^{\mathbf{X}}$-almost all
$\mathbf{x}\in\mathbb{R}^{m}$, whereby as usual $\mathbb{P}^{\mathbf{X}}$
denotes the push-forward of $\mathbb{P}$ via $\mathbf{X}$. In case
$(\mathbf{X},\mathbf{Y})$ has $C\in\mathcal{C}^{d}$ as distribution function
(restricted to $\mathbb{I}^{d}$) we let
$K_{C}:\mathbb{I}^{m}\times\mathcal{B}(\mathbb{I}^{d-m})\to\mathbb{I}$ denote
(a version of) the regular conditional distribution of $\mathbf{Y}$ given
$\mathbf{X}$ and simply refer to it as $m$-_Markov kernel_ of $C$. Defining
the $\mathbf{x}$-section $G_{\mathbf{x}}$ of a set
$G\in\mathcal{B}(\mathbb{I}^{d})$ w.r.t. the first $m$ coordinates by
$G_{\mathbf{x}}:=\\{\mathbf{y}\in\mathbb{I}^{d-m}:(\mathbf{x},\mathbf{y})\in
G\\}\in\mathcal{B}(\mathbb{I}^{d-m})$ the well-known _disintegration theorem_
implies

$\displaystyle\mu_{C}(G)=\int_{\mathbb{I}^{m}}K_{C}(\mathbf{x},G_{\mathbf{x}})\,\mathrm{d}\mu_{C^{1:m}}(\mathbf{x}).$

It is well-known that the disintegration theorem also holds for general finite
measures in which case the conditional measures $K(\cdot,\cdot)$ are not
necessarily probability measures but general finite measures (sub- or super
Markov kernels). For more background on conditional expectation and
disintegration we refer to the [11] and [17].

An Archimedean _generator_ $\psi$ is a continuous, non-increasing function
$\psi:[0,\infty)\to[0,1]$ fulfilling $\psi(0)=1$,
$\lim_{z\to\infty}\psi(z)=0=:\psi(\infty)$ and being strictly decreasing on
$[0,\inf\\{z\in[0,\infty]:\psi(z)=0\\}]$ (with the convention
$\inf\emptyset:=\infty$). For every Archimedean generator $\psi$ we will let
$\varphi:[0,1]\rightarrow[0,\infty]$ denote its pseudo-inverse defined by
$\varphi(y)=\inf\\{z\in[0,\infty]:\psi(z)=y\\}$ for every $y\in[0,1]$.
Obviously $\varphi$ is strictly decreasing on $[0,1]$ and fulfills
$\varphi(1)=0$, moreover it is straightforward to verify that $\varphi$ is
right-continuous at $0$ (for a short discussion of this property see Section 4
in [14]). If $\varphi(0)=\infty$ we refer to $\psi$ (and $\varphi$) as
_strict_ and as _non-strict_ otherwise. A copula $C\in\mathcal{C}^{d}$ is
called _Archimedean_ (in which case we write
$C\in\mathcal{C}_{\text{ar}}^{d}$) if there exists some Archimedean generator
$\psi$ with

$\displaystyle
C_{\psi}(\mathbf{x})=\psi(\varphi(x_{1})+\cdots+\varphi(x_{d}))$

for every $\mathbf{x}\in\mathbb{I}^{d}$. Following [21],
$C_{\psi}(\mathbf{x})=\psi(\varphi(x_{1})+\cdots+\varphi(x_{d}))$ is a
$d$-dimensional copula if, and only if, $\psi$ is a $d$-monotone Archimedean
generator on $[0,\infty)$, i.e., if, and only if, $\psi$ is an Archimedean
generator fulfilling that $(-1)^{d-2}\psi^{(d-2)}$ exists on $(0,\infty)$, is
non-negative, non-increasing and convex on $(0,\infty)$, whereby, as usual,
$g^{(m)}$ denotes the $m$-th derivative of a function $g$. Moreover (again see
[21]) it is straightforward to verify that in the latter case
$(-1)^{m}\psi^{(m)}$ exists on $(0,\infty)$, is non-negative, non-increasing
and convex on $(0,\infty)$ for every $m\in\\{0,\ldots,d-2\\}$. In the sequel
we will sometimes simply write $C$ instead of $C_{\psi}$ when no confusion may
arise. Furthermore in the sequel we will simply refer to Archimedean
generators as ‘generators’.

Letting $D^{-}g$ and $D^{+}g$ denote the left- and right- hand derivative of a
function $g$, respectively, convexity of $(-1)^{d-2}\psi^{(d-2)}$ implies that
both, $D^{-}\psi^{(d-2)}(z)$ and $D^{+}\psi^{(d-2)}(z)$ exist for every
$z\in(0,\infty)$ and that the two derivatives coincide outside an at most
countable set (see [12, 23]) \- in fact, for every continuity point $z$ of
$D^{-}\psi^{(d-2)}$ we have $D^{-}\psi^{(d-2)}(z)=D^{+}\psi^{(d-2)}(z)$.
Moreover, every $d$-monotone generator $\psi$ fulfills
$\lim_{z\rightarrow\infty}\psi^{(m)}(z)=0$ for every $m\in\\{0,\ldots,d-2\\}$
as well as $\lim_{z\rightarrow\infty}D^{-}\psi^{(d-2)}(z)=0$. Indeed,
$\lim_{z\rightarrow\infty}\psi^{\prime}(z)=0$ directly follows from
monotonicity and convexity of $\psi$ since $d$-monotonicity implies that
$-\psi^{\prime}$ is decreasing and convex too; proceeding iteratively yields
the assertion. Notice that Lemma 5.3 yields an even simpler direct proof of
this assertion. In the sequel we will also use the fact that (again by
convexity, see [12, 23]) we can reconstruct the generator $\psi$ from its
derivatives in the sense that ($m\in\\{1,\ldots,d-2\\}$)

(2.1) $\displaystyle\psi^{(m-1)}(z)$
$\displaystyle=\int_{[z,\infty)}-\psi^{(m)}(s)\
\mathrm{d}\lambda(s),\quad\psi^{(d-2)}(z)=\int_{[z,\infty)}-D^{-}\psi^{(d-2)}(s)\
\mathrm{d}\lambda(s).$

In order to have a one-to-one correspondence between copulas and their
generator we follow [14] and from now on implicitly assume that all generators
are normalized in the sense that $\varphi(\frac{1}{2})=1$, or equivalently,
$\psi(1)=\frac{1}{2}$ holds.

According to [21], an Archimedean copula $C\in\mathcal{C}_{\text{ar}}^{d}$ is
absolutely continuous if, and only if, $\psi^{(d-1)}$ exists and is absolutely
continuous on $(0,\infty)$. In this case a version of the density $c$ of $C$
is given by

(2.2) $\displaystyle
c(\mathbf{x})=\mathbf{1}_{(0,1)^{d}}(\mathbf{x})\prod_{i=1}^{d}\varphi^{\prime}(x_{i})\cdot
D^{-}\psi^{(d-1)}\big{(}\varphi(x_{1})+\cdots+\varphi(x_{d})\big{)}.$

In the sequel we will use the handy consequence that lower dimensional
marginals of $d$-dimensional Archimedean copulas are absolutely continuous
([21, Proposition 4.1]).

##  3\. Markov kernel, mass distribution and Kendall distribution function of
multivariate Archimedean copulas

In the proceedings contribution [15] the authors derive an explicit expression
for (a version of) the $(d-1)$-Markov kernel of $d$-variate Archimedean
copulas which, in turn, allows to derive the well-known formulas for level-set
mass and the Kendall distribution function of $d$-variate Archimedean copulas
in an alternative way. Considering that Markov kernels are key for the rest of
this paper we introduce a (slightly modified) version of the Markov kernel
considered in [15] and prove its properties in detail. Furthermore, we restate
the already known formulas for the Kendall distribution function and level-set
mass, add a new geometric interpretation, and, for the sake of completeness,
include the corresponding purely Markov kernel-based proofs in the Appendix.

For every $t\in(0,1]$ define the $t$-level set of
$C\in\mathcal{C}_{\text{ar}}^{d}$ by

(3.1)
$L_{t}:=\left\\{(\mathbf{x},y)\in\mathbb{I}^{d-1}\times\mathbb{I}:C(\mathbf{x},y)=t\right\\}=\left\\{(\mathbf{x},y)\in\mathbb{I}^{d-1}\times\mathbb{I}:\sum_{i=1}^{d-1}\varphi(x_{i})+\varphi(y)=\varphi(t)\right\\}$

and for $t=0$ set

(3.2)
$L_{0}:=\left\\{(\mathbf{x},y)\in\mathbb{I}^{d-1}\times\mathbb{I}:C(\mathbf{x},y)=0\right\\}=\left\\{(\mathbf{x},y)\in\mathbb{I}^{d-1}\times\mathbb{I}:\sum_{i=1}^{d-1}\varphi(x_{i})+\varphi(y)\geq\varphi(0)\right\\}.$

Subsequently we will work with the level sets of the $(d-1)$-dimensional
marginal of $C$ defined analogously and denote them by $L_{t}^{1:d-1}$ and
$L_{0}^{1:d-1}$, respectively. As in the bivariate setting (see [14]) we can
define functions $f^{t}$ whose graph coincides with $L^{t}$ for $t\in(0,1]$
and with the boundary of $L^{0}$ for $t=0$: In fact, for $t=0$ defining the
function $f^{0}:\mathbb{I}^{d-1}\to\mathbb{I}$ by

$\displaystyle f^{0}(\mathbf{x})=\begin{cases}1&\text{ if }\mathbf{x}\in
L_{0}^{1:d-1}\\\
\psi\left(\varphi(0)-\sum_{i=1}^{d-1}\varphi(x_{i})\right)&\text{ if
}\mathbf{x}\not\in L_{0}^{1:d-1}\end{cases}$

with the conventions $\psi(\infty)=0$ as well as $\psi(u)=1$ for all $u<0$ and
for $t\in(0,1]$, defining the upper $t$-cut of $C^{1:d-1}$ by
$[C^{1:d-1}]_{t}=\\{\mathbf{x}\in\mathbb{I}^{d-1}:C^{1:d-1}(\mathbf{x})\geq
t\\}$ and considering the function $f^{t}:[C^{1:d-1}]_{t}\to\mathbb{I}$ given
by

$\displaystyle
f^{t}(\mathbf{x}):=\psi\left(\varphi(t)-\sum_{i=1}^{d-1}\varphi(x_{i})\right).$

yields the above-mentioned property. It is straightforward to verify that for
$\mathbf{x}\not\in$ $L_{0}^{1:d-1}$ and $y<f^{0}(\mathbf{x})$ we have
$(\mathbf{x},y)\in L_{0}$ and that for strict Archimedean copulas
$\mathbf{x}\in L_{0}^{1:d-1}$ if, and only if, $M(\mathbf{x})=0$ where $M$
denotes the $d$-dimensional minimum copula.

######  Theorem 3.1.

Suppose that $C\in\mathcal{C}_{\text{ar}}^{d}$ has generator $\psi$. Then
setting

(3.3) $\displaystyle
K_{C}(\mathbf{x},[0,y]):=\begin{cases}1,&M(\mathbf{x})=1\emph{ or
}\mathbf{x}\in L_{0}^{1:d-1}\\\ 0,&M(\mathbf{x})<1,\mathbf{x}\not\in
L_{0}^{1:d-1},y<f^{0}(\mathbf{x})\\\
\frac{D^{-}\psi^{(d-2)}\left(\sum\limits_{i=1}^{d-1}\varphi(x_{i})+\varphi(y)\right)}{D^{-}\psi^{(d-2)}\left(\sum\limits_{i=1}^{d-1}\varphi(x_{i})\right)},&M(\mathbf{x})<1,\mathbf{x}\not\in
L_{0}^{1:d-1},y\geq f^{0}(\mathbf{x})\end{cases}$

yields (a version of) the $(d-1)$-Markov kernel of $C$.

###### Proof.

First notice that absolute continuity of $C^{1:d-1}$ implies

$\displaystyle 0=\mu_{C^{1:d-1}}(L_{0}^{1:d-1})$
$\displaystyle=\mu_{C}(L_{0}^{{}^{:}d-1}\times\mathbb{I}).$

so $\mu_{C^{1:d-1}}(L_{0}^{1:d-1}\cup M^{-1}(\\{1\\}))=0$, so the first
condition on the right hand side of eq. (3.3) only holds for a set of
$\mu_{C^{1:d-1}}$-measure $0$.

We start by showing that $K_{C}$ defined according to eq. (3.3) is indeed a
$(d-1)$-Markov kernel and then prove that it is a Markov kernel of $C$. Fix
$y\in\mathbb{I}$. Since measurability of $f^{0}$ and $D^{-}\psi^{(d-2)}$ are a
direct consequence of the properties of $\psi$, using continuity of $\varphi$
yields measurability of the mapping $\mathbf{x}\mapsto
K_{C}(\mathbf{x},[0,y])$. Building upon that, applying using a standard Dynkin
System argument (see [4, Theorem 2]) yields measurability of
$\mathbf{x}\mapsto K_{C}(\mathbf{x},E)$ for every Borel set
$E\in\mathcal{B}(\mathbb{I})$.
As for $\mathbf{x}=1$ or fixed $\mathbf{x}\in L_{0}^{1:d-1}$ the map $y\mapsto
K_{C}(\mathbf{x},[0,y])$ is obviously a univariate distribution function
consider $\mathbf{x}\not\in L_{0}^{1:d-1}$. Then monotonicity and left-
continuity of $(-1)^{d-2}D^{-}\psi^{(d-2)}$ implies that $y\mapsto
K_{C}(\mathbf{x},[0,y])$ is increasing and right-continuous. Moreover we
obviously have $K_{C}(\mathbf{x},[0,1])=1$, so $y\mapsto
K_{C}(\mathbf{x},[0,y])$ is a univariate distribution function and it remains
to show that it is a $(d-1)$-Markov kernel of $C$, i.e., that

(3.4) $\displaystyle C(\mathbf{x},y)$
$\displaystyle=\int_{[\mathbf{0},\mathbf{x}]}K_{C}(\mathbf{s},[0,y])\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{s})$

holds for all $\mathbf{x}\in\mathbb{I}^{d-1}$ and every $y\in\mathbb{I}$.
The case $y=1$ is trivial and for $y=0$ we have that
$K_{C}(\mathbf{x},\\{0\\})=0$ for $\mu_{C^{1:d-1}}$-almost every $\mathbf{x}$,
implying that eq. (3.4) holds. For $y\in(0,1)$ considering that
$L_{0}^{1:d-1}$ is a $\mu_{C^{1:d-1}}$-null set and using absolute continuity
of $\mu_{C^{1:d-1}}$ yields ($\Upsilon:=\mathbb{I}^{d-1}\setminus
L_{0}^{1:d-1}$)

$\displaystyle I$
$\displaystyle:=\int_{[\mathbf{0},\mathbf{x}]\cap\Upsilon}K_{C}(\mathbf{s},[0,y])\
\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{s})=\int_{[\mathbf{0},\mathbf{x}]\cap\Upsilon}K_{C}(\mathbf{s},[0,y])c^{1:d-1}(\mathbf{s})\
\mathrm{d}\lambda(\mathbf{s})$
$\displaystyle=\int_{\Upsilon\cap\\{\mathbf{t}\in(0,1)^{d-1}:y\geq
f^{0}(\mathbf{t})\\}\cap[\mathbf{0},\mathbf{x}]}\tfrac{D^{-}\psi^{(d-2)}(\sum_{i=1}^{d-1}\varphi(s_{i})+\varphi(y))}{D^{-}\psi^{(d-2)}(\sum_{i=1}^{d-1}\varphi(s_{i}))}\prod_{i=1}^{d-1}\varphi^{\prime}(s_{i})D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(s_{i})\right)\
\mathrm{d}\lambda(\mathbf{s})$
$\displaystyle=\int_{\\{\mathbf{t}\in(0,1)^{d-1}\setminus
L_{0}^{1:d-1}:\,y\geq
f^{0}(\mathbf{t})\\}\cap[\mathbf{0},\mathbf{x}]}D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(s_{i})+\varphi(y)\right)\cdot\prod_{i=1}^{d-1}\varphi^{\prime}(s_{i})\
\mathrm{d}\lambda(\mathbf{s}).$

Notice that on the one hand, for $\mathbf{s}\not\in L_{0}^{1:d-1}$ we have
$y<f^{0}(\mathbf{s})$ if, and only if,
$\sum_{i=1}^{d-1}\varphi(s_{i})+\varphi(y)>\varphi(0)$, implying
$D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(s_{i})+\varphi(y)\right)=0$,
and, on the other hand, $\mathbf{s}\in L_{0}^{1:d-1}$ also yields
$D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(s_{i})+\varphi(y)\right)=0$.
Therefore in the strict and the non-strict case we get

$\displaystyle I$ $\displaystyle=\int_{(\mathbf{0},\mathbf{x}]\setminus
L_{0}^{1:d-1}}D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(s_{i})+\varphi(y)\right)\cdot\prod_{i=1}^{d-1}\varphi^{\prime}(s_{i})\
\mathrm{d}\lambda(\mathbf{s})$
$\displaystyle=\int_{(\mathbf{0},\mathbf{x}]\cap(0,1)^{d-1}}D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(s_{i})+\varphi(y)\right)\cdot\prod_{i=1}^{d-1}\varphi^{\prime}(s_{i})\
\mathrm{d}\lambda(\mathbf{s})$
$\displaystyle=\int_{(\mathbf{0},\mathbf{x}_{1:d-2}]}\prod_{i=1}^{d-2}\varphi^{\prime}(s_{i})\int_{(0,x_{d-1}]}\varphi^{\prime}(s_{d-2})D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(s_{i})+\varphi(y)\right)\mathrm{d}\lambda(s_{d-1})\mathrm{d}\lambda(\mathbf{s}_{1:d-2}).$

Hence, using change of coordinates together with the fact that
$\psi^{(m)}(\infty)=0$ holds for $m=1,2,\ldots,d-2$ it follows that

$\displaystyle I$
$\displaystyle=\int_{(\mathbf{0},\mathbf{x}_{1:d-2}]}\prod_{i=1}^{d-2}\varphi^{\prime}(s_{i})\lim_{\Delta\to
0}\Biggl{\\{}\psi^{(d-2)}\left(\sum_{i=1}^{d-2}\varphi(s_{i})+\varphi(x_{d-1})+\varphi(y)\right)$
$\displaystyle\qquad-\psi^{(d-2)}\left(\sum_{i=1}^{d-2}\varphi(s_{i})+\varphi(\Delta)+\varphi(y)\right)\Biggr{\\}}\mathrm{d}\lambda(\mathbf{s}_{1:d-2})$
$\displaystyle=\int_{(\mathbf{0},\mathbf{x}_{1:d-2}]}\prod_{i=1}^{d-2}\varphi^{\prime}(s_{i})\psi^{(d-2)}\left(\sum_{i=1}^{d-2}\varphi(s_{i})+\varphi(x_{d-1})+\varphi(y)\right)\mathrm{d}\lambda(\mathbf{s}_{1:d-2}).$

Proceeding analogously $d-3$ times finally yields

$\displaystyle I$
$\displaystyle=\int_{(0,x_{1}]}\varphi^{\prime}(s_{1})\psi^{\prime}\left(\varphi(s_{1})+\varphi(x_{2})+\cdots+\varphi(x_{d-1})+\varphi(y)\right)\mathrm{d}\lambda(s_{1})$
$\displaystyle=\psi\left(\sum_{i=1}^{d-1}\varphi(x_{i})+\varphi(y)\right)=C(\mathbf{x},y)$

as desired. ∎

######  Remark 3.2.

As mentioned in the proof above $\mu_{C^{1:d-1}}(L_{0}^{1:d-1})=0$, implying
that in the first line in equation (3.3) we could substitute the univariate
distribution function $y\mapsto 1$ by any other univariate distribution
function $F$ fulfilling $F(1)=1$.

######  Remark 3.3.

It also seems feasible to consider $m$-kernels for some
$m\in\\{2,\ldots,d-2\\}$ instead of $(d-1)$-kernels, i.e., to condition on $m$
instead of $d-1$ coordinates. As shown by the subsequent results, however,
although $d-1$-kernels involve the highest derivatives of the generator they
are easy to handle and provide various new results, particularly with respect
to the interplay with the Williamson transform as discussed in Section 5.

Utilizing Theorem 3.1 allows to derive the well-known formulas ([21]) for the
level set masses and the Kendall distribution function of multivariate
Archimedean copulas easily via Markov kernels and disintegration (see
Propositions A.1 and A.2 in the Appendix). In fact, for $t\in(0,1]$ the
identity

(3.5)
$\displaystyle\mu_{C}(L_{t})=\frac{(-\varphi(t))^{d-1}}{(d-1)!}\cdot\big{(}D^{-}\psi^{(d-2)}(\varphi(t))-D^{-}\psi^{(d-2)}(\varphi(t-))\big{)}.$

can be shown. Moreover, if $C$ is strict then $\mu_{C}(L_{0})=0$ and for non-
strict $C$

(3.6) $\displaystyle\mu_{C}(L_{0})$
$\displaystyle=\frac{(-\varphi(0))^{d-1}}{(d-1)!}\cdot
D^{-}\psi^{(d-2)}(\varphi(0))$

holds. Letting $F_{K}^{d}$ denote the Kendall distribution function of $C$,
for $t>0$ we have

(3.7) $\displaystyle
F_{K}^{d}(t)=D^{-}\psi^{(d-2)}(\varphi(t))\frac{(-1)^{d-1}}{(d-1)!}\varphi(t)^{d-1}+\sum_{k=0}^{d-2}\psi^{(k)}(\varphi(t))\frac{(-1)^{k}}{k!}\varphi(t)^{k}.$

Moreover for $t=0$ and strict $C$ we have $F_{K}^{d}(0)=0$, and for non-strict
$C$

(3.8) $\displaystyle
F_{K}^{d}(0)=D^{-}\psi^{(d-2)}(\varphi(0))\cdot\frac{(-1)^{d-1}}{(d-1)!}\cdot\varphi(0)^{d-1}.$

holds.

######  Remark 3.4.

In the bivariate setting it is well-known that the formula for the level set
masses can nicely be interpreted geometrically (see [4, 22]). In fact,
following [4], given a discontinuity $b$ of $D^{+}\varphi$ we have

$\displaystyle\mu_{C}(L_{b})=\varphi(b)\cdot\left(\frac{1}{D^{-}\varphi(b)}-\frac{1}{D^{+}\varphi(b)}\right),$

i.e., $\mu_{C}(L_{b})$ corresponds to the length of the line segment on the
$x$-axis generated by the left-hand and right-hand tangents of $\varphi$ at
$b$ (see Figure 1 in [4]). Translating from $\varphi$ and $x$-axis to $\psi$
and $y$-axis yields as special case of eq. (3.5)

$\displaystyle\mu_{C}(L_{b})=\varphi(b)\cdot\left(D^{-}\psi(\varphi(b))-D^{-}\psi(\varphi(b-))\right).$

To establish the multivariate geometric analogue, rather than tangent lines we
can consider the left and right hand Taylor polynomials of $\psi$ of order
$d-1$ at $a\in[0,\infty)$, i.e.,

$\displaystyle
T_{d-1}^{\pm}\psi(z,a):=D^{\pm}\psi^{(d-2)}(a)\cdot\frac{(z-a)^{d-1}}{(d-1)!}+\sum_{k=0}^{d-2}\psi^{(k)}(a)\cdot\frac{(z-a)^{k}}{k!}.$

Having that yields

$T^{-}_{d-1}\psi(0,\varphi(t))=D^{-}\psi^{(d-2)}(\varphi(t))\frac{(-1)^{d-1}\varphi(t)^{d-1}}{(d-1)!}+\sum_{k=0}^{d-2}\psi^{(k)}(\varphi(t))\frac{(-1)^{k}\varphi(t)^{k}}{k!}=F_{K}^{d}(t)$

and (3.5) reduces to

$\displaystyle\mu_{C}(L_{t})=T_{d-1}^{-}\psi(0,\varphi(t))-T_{d-1}^{+}\psi(0,\varphi(t))=F_{K}^{d}(t)-F_{K}^{d}(t-),$

i.e., the $t$-level set mass corresponds to the difference of $y$-intercepts
of the left and right hand Taylor polynomials of $\psi$ of order $d-1$ at
$\varphi(t)$. Panel 2 of Figure 1 in Example 3.5 illustrates this
interpretation for $d=3$.

######  Example 3.5.

We construct an Archimedean generator $\psi$ inducing some
$C_{\psi}\in\mathcal{C}_{\text{ar}}^{3}$ with $1=z_{0}$ being a discontinuity
point of $(D^{-}\psi^{\prime})$ and provide a geometric interpretation of
$\mu_{C}(L_{z_{0}})$ in terms of $\psi$. As depicted in the left panel of
Figure 1 we start with some non-negative, convex and decreasing function
$(-1)\tilde{\psi}^{(1)}$ (gray) with $D^{+}\tilde{\psi}^{(1)}(z_{0})\neq
D^{-}\tilde{\psi}^{(1)}(z_{0})$. Considering

$\displaystyle\tilde{\psi}(z):=\int_{[z,\infty)}(-1)\tilde{\psi}^{(1)}(s)\
\mathrm{d}\lambda(s)$

as well as setting $\psi(z)=\tilde{\psi}(z)/\tilde{\psi}(0)$ yields a
generator of a three-dimensional Archimedean copula $C_{\psi}$ (gray curve in
the right panel). The length of the vertical red line segment formed by the
left and right hand Taylor polynomials of order $2$ at $z_{0}$ (right panel)
coincides with $\mu_{C}(L_{z_{0}})$.

Figure 1. Stepwise construction of a non-strict $3$-monotone Archimedean
generator and geometric interpretation of the $z_{0}$-level set mass in terms
of the vertical red line segment formed by the two Taylor parabolas in $z_{0}$
as considered in Example 3.5.

##  4\. Characterizing pointwise convergence in $\mathcal{C}^{d}_{ar}$ and the
interrelation with weak conditional convergence

In [2, Section 6] it was shown that within the class of bivariate Archimedean
copulas pointwise convergence and weak conditional convergence (defined as
weak conditional convergence of $\lambda$-almost all conditional
distributions) are equivalent. Moreover, it was shown that in the general
bivariate setting, weak conditional convergence implies pointwise convergence
not necessarily vice versa. We now tackle the question whether analogous
results hold in the general $d\geq 3$-dimensional setting. Considering that -
contrary to the bivariate case $d=2$ \- for two copulas
$A,B\in\mathcal{C}^{d}$ we do not necessarily have $A^{1:d-1}=B^{1:d-1}$ we
first need to discuss potential extensions of the notion of weak conditional
convergence to $\mathcal{C}^{d}$.

The seemingly most natural approach would be to say that a sequence
$(A_{n})_{n\in\mathbb{N}}$ of $d$-dimensional copulas converges _weakly
conditional_ to $A\in\mathcal{C}^{d}$ if, and only if, there exists a set
$\Lambda$ with $\mu_{A^{1:d-1}}(\Lambda)=1$ such that for every
$\mathbf{x}\in\Lambda$ the sequence
$\left(K_{A_{n}}(\mathbf{x},\cdot)\right)_{n\in\mathbb{N}}$ of probability
measures on $\mathcal{B}(\mathbb{I})$ converges to the probability measure
$K_{A}(\mathbf{x},\cdot)$. As pointed out in [9], however, for
$A,B\in\mathcal{C}^{d}$ we might even have that $A^{1:d-1},B^{1:d-1}$ (or,
more precisely, the measures $\mu_{A^{1:d-1}},\mu_{B^{1:d-1}}$) are singular
with respect to each other, making it unreasonable to compare
$K_{A}(\mathbf{x},\cdot)$ and $K_{B}(\mathbf{x},\cdot)$ since they are only
defined uniquely $\mu_{A^{1:d-1}}$-a.e. and $\mu_{B^{1:d-1}}$-a.e.,
respectively (see [9, Example 4.10] for an illustration of this scenario).

As a consequence, the afore-mentioned natural concept of weak conditional
convergence does not yield a reasonable notion in full generality. For certain
families of copulas such as classes of copulas with identical marginals, or
for Archimedean copulas, however, considering weak conditional convergence
does make sense. In fact, given $C_{1},C_{2}\in\mathcal{C}_{\text{ar}}^{d}$ we
already know that $C_{1}^{1:d-1}$ and $C_{2}^{1:d-1}$ are both not only
absolutely continuous but according to eq. (2.2) the corresponding densities
$c_{1}^{1:d-1},c_{2}^{1:d-1}$ fulfill $c_{i}^{1:d-1}(\mathbf{x})>0$ for every
$\mathbf{x}$ outside of the respective zero set $L_{0}^{1:d-1}$, so it can not
happen that $C_{1}^{1:d-1}$ and $C_{2}^{1:d-1}$ are singular w.r.t. each
other.

In what follows we will therefore work with the afore-mentioned notion of weak
conditional convergence in $\mathcal{C}^{d}_{ar}$ and prove the following main
result of this section in several steps:

######  Theorem 4.1.

Suppose that $C,C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean copulas
with generators $\psi,\psi_{1},\psi_{2},\ldots$, respectively. Then the
following assertions are equivalent (_Cont_ $(g)$ denotes the set of
continuity points of a function $g$):

  1. (1)

$(C_{n})_{n\in\mathbb{N}}$ converges uniformly to $C$.

  2. (2)

$(C_{n}^{i:j})_{n\in\mathbb{N}}$ converges uniformly to $C^{i:j}$ for all
$i,j\in\\{1,\ldots,d\\}$ with $i\neq j$.

  3. (3)

$(\varphi_{n})_{n\in\mathbb{N}}$ converges pointwise to $\varphi$ on $(0,1]$.

  4. (4)

$(\psi_{n})_{n\in\mathbb{N}}$ converges uniformly to $\psi$ on $[0,\infty)$.

  5. (5)

$(\psi_{n}^{(m)})_{n\in\mathbb{N}}$ converges pointwise to $\psi^{(m)}$ on
$(0,\infty)$ for every $m\in\\{1,2,\ldots,d-2\\}$ and
$(D^{-}\psi_{n}^{(d-2)})_{n\in\mathbb{N}}$ converges pointwise to
$D^{-}\psi^{(d-2)}$ on _Cont_ $(D^{-}\psi^{(d-2)})$.

  6. (6)

$(c_{n}^{1:m})_{n\in\mathbb{N}}$ converges to $c^{1:m}$ $\lambda_{d}$-almost
everywhere in $\mathbb{I}^{m}$ for every $m\in\\{2,3,\ldots,d-1\\}$.

Furthermore, any of the six assertions implies weak conditional convergence of
$(C_{n})_{n\in\mathbb{N}}$ to $C$.

Obviously, assuming uniform convergence of $(C_{n})_{n\in\mathbb{N}}$ to $C$
yields convergence of all marginal copulas and, in particular, of the
bivariate marginals. Thus, the equivalence of (1), (2) and (3) follows
directly from the results in the two-dimensional setting established in [14].

######  Remark 4.2.

Theorem 4.1 constitutes the natural extension of [14, Theorem 4.2]
characterizing uniform convergence within the space of bivariate Archimedean
copulas. In the multivariate setting, however, the afore-mentioned notion of
weak conditional convergence in $\mathcal{C}^{d}_{ar}$ is a consequence of
rather than an equivalence to uniform convergence. Slightly modifying the
notion and incorporating the marginal densities, however, an equivalence can
be established, see point (5) in Theorem 4.9 at the end of this section.

In what follows, we show the equivalence of the assertions (4), (5) and (6)
and then conclude the section by deriving weak conditional convergence in
several steps. We start with the following lemma clarifying the relationship
between convergence of the generators and their pseudo-inverses. The proof of
an analogous result in the context of t-norms with multiplicative generators
can be found in [16, Theorem 8.14].

######  Lemma 4.3.

Suppose that $\psi,\psi_{1},\psi_{2},\ldots$ are Archimedean generators with
pseudo-inverses $\varphi,\varphi_{1},\varphi_{2},\ldots$, respectively. Then
$(\varphi_{n})_{n\in\mathbb{N}}$ converges pointwise to $\varphi$ on $(0,1]$
if, and only if, $(\psi_{n})_{n\in\mathbb{N}}$ converges to $\psi$ uniformly
on $[0,\infty)$.

###### Proof.

Defining

$\displaystyle\tilde{\psi}(w):=\begin{cases}\psi(-w)&\text{ if }w\leq 0\\\
1&\text{ otherwise}\end{cases}$

yields a univariate distribution function $\tilde{\psi}$ on $\mathbb{R}$ whose
pseudo-inverse $(\tilde{\psi})^{-}:(0,1]\rightarrow\mathbb{R}$ coincides with
$-\varphi$. Considering that weak convergence of distribution functions is
equivalent to weak convergence of their pseudo-inverses (see [26, Lemma
21.2]), using continuity of the involved functions it directly follows that
pointwise convergence of $\tilde{\psi}_{n}$ to $\tilde{\psi}$ pointwise on
$\mathbb{R}$ is equivalent to pointwise convergence of $-\varphi_{n}$ to
$-\varphi$ on $(0,1]$ for $n\to\infty$. In other words: $\psi_{n}\to\psi$
pointwise on $[0,\infty)$ if, and only if, $\varphi_{n}\to\varphi$ pointwise
on $(0,1]$. Finally, uniform convergence of $\tilde{\psi}_{n}$ to
$\tilde{\psi}$ on $\mathbb{R}$ is a direct consequence of the fact that
pointwise convergence of a sequence of univariate distribution functions to a
continuous distribution function implies uniform convergence. Having that,
obviously uniform convergence of $(\psi_{n})_{n\in\mathbb{N}}$ to $\psi$ on
$[0,\infty)$ follows. ∎

The next lemma focuses on convergence of the derivatives of the Archimedean
generators. Thereby we say that a sequence $(f_{n})_{n\in\mathbb{N}}$ of real
functions $(f_{n})_{n\in\mathbb{N}}$ _converges continuously_ to $f$ if for
every sequence $(x_{n})_{n\in\mathbb{N}}$ converging to $x\in$ Cont$(f)$ we
have $\lim_{n\to\infty}f_{n}(x_{n})=f(x)$ (cf. [18]).

######  Lemma 4.4.

Suppose that $C,C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean copulas
with generators $\psi,\psi_{1},\psi_{2},\ldots$, respectively. If
$(C_{n})_{n\in\mathbb{N}}$ converges pointwise to $C$ then for every
$m\in\\{0,1,2,\ldots,d-2\\}$ we have

$\displaystyle\lim_{n\to\infty}\psi^{(m)}_{n}(z)=\psi^{(m)}(z).$

for every $z\in(0,\infty)$ as well as
$\lim_{n\to\infty}D^{-}\psi^{(d-2)}_{n}(z)=D^{-}\psi^{(d-2)}(z)$ for every
$z\in$ _Cont_ $(D^{-}\psi^{(d-2)})$. Moreover, in both situations the
convergence is continuous, i.e., for every sequence $(z_{n})_{n\in\mathbb{N}}$
in $(0,\infty)$ converging to $z\in\emph{Cont}(D^{-}\psi^{(d-2)})$ it holds
that
$\lim_{n\rightarrow\infty}D^{-}\psi_{n}^{(d-2)}(z_{n})=D^{-}\psi^{(d-2)}(z)$.
Vice versa, if $(\psi_{n}^{(m)})_{n\in\mathbb{N}}$ converges pointwise to
$\psi^{(m)}$ on $(0,\infty)$ for some $m\in\\{1,2,\ldots,d-2\\}$ then
$(C_{n})_{n\in\mathbb{N}}$ converges pointwise to $C$ and the same implication
holds if $\lim_{n\to\infty}D^{-}\psi^{(d-2)}_{n}(z)=D^{-}\psi^{(d-2)}(z)$ for
every $z\in$ _Cont_ $(D^{-}\psi^{(d-2)})$.

###### Proof.

We already know that pointwise convergence of $(C_{n})_{n\in\mathbb{N}}$ to
$C$ is equivalent to uniform convergence of $(\psi_{n})_{n\in\mathbb{N}}$ to
$\psi$ on $[0,\infty)$. Convexity of generators implies that the sequence
$(\psi_{n}^{\prime})_{n\in\mathbb{N}}$ converges pointwise to $\psi^{\prime}$
on Cont$(\psi^{\prime})=(0,\infty)$ (see, e.g., [24]). Considering that
$(-1)^{m}\psi^{(m)},(-1)^{m}\psi_{1}^{(m)},(-1)^{m}\psi_{2}^{(m)},\ldots$ are
convex functions for every $m\in\\{1,\ldots,d-2\\}$ too, applying [24],
continuous convergence follows. The fact that
$(D^{-}\psi_{n}^{(d-2)})_{n\in\mathbb{N}}$ converges continuously to
$D^{-}\psi^{(d-2)}$ on Cont$(D^{-}\psi^{(d-2)})$ is a consequence of Lemma A.3
in the Appendix.
We prove the reverse implication only for the case $d=3$ and assume that
$(D^{-}\psi_{n}^{(d-2)})_{n\in\mathbb{N}}$ converges to $D^{-}\psi^{(d-2)}$ on
Cont$(D^{-}\psi^{(d-2)})$ since the analogous statement for
$m\in\\{1,\ldots,d-2\\}$ follows in the same manner and the extension to
arbitrary $d\geq 3$ is obvious. For $z\in[0,\infty)$ according to eq. (2.1) we
have

$\displaystyle\psi(z)$
$\displaystyle=\int_{[z,\infty)}-\psi^{\prime}(s)\mathrm{d}\lambda(s)=\int_{[z,\infty)}-\int_{[s,\infty)}-D^{-}\psi^{\prime}(t)\mathrm{d}\lambda(t)\mathrm{d}\lambda(s)$
$\displaystyle=\int_{[z,\infty)}\int_{[s,\infty)}\underbrace{D^{-}\psi^{\prime}(t)}_{\geq
0}\mathrm{d}\lambda(t)\mathrm{d}\lambda(s)$

and the same holds for every $\psi_{n}$. Using $\psi(0)=\psi_{n}(0)=1$ we can
interpret the functions $\iota_{n},\iota:\Delta\rightarrow[0,\infty)$, defined
by

$\displaystyle\iota_{n}(t,s):=D^{-}\psi_{n}^{\prime}(t),\quad\iota(t,s):=D^{-}\psi^{\prime}(t)$

as probability densities on the measure space
$(\Delta,\mathcal{B}(\Delta),\lambda_{2})$ with $\Delta$ denoting the closed
set $\Delta:=\\{(x,y)\in[0,\infty)^{2}:y\geq x\\}$. By assumption, the
sequence $(\iota_{n})_{n\in\mathbb{N}}$ converges $\lambda_{2}$-almost
everywhere on $\Delta$ to $\iota$, so applying Scheffe’s theorem (or Riesz’
theorem, see [19]) yields

$\displaystyle\lim_{n\rightarrow\infty}\int_{\Delta}|\iota_{n}(t,s)-\iota(t,s)|d\lambda_{2}(t,s)=0.$

Hence, for an arbitrary $z\in(0,\infty)$ using the triangle inequality it
follows that

$\displaystyle|\psi_{n}(z)-\psi(z)|$
$\displaystyle=\left|\int_{[z,\infty)}\int_{[s,\infty]}D^{-}\psi_{n}^{\prime}(t)-D^{-}\psi^{\prime}(t)\,\mathrm{d}\lambda(t)\mathrm{d}\lambda(s)\right|$
$\displaystyle\leq\int_{\Delta\cap[z,\infty)^{2}}|\iota_{n}(t,s)-\iota(t,s)|\mathrm{d}\lambda_{2}(t,s)\leq\int_{\Delta}|\iota_{n}-\iota|\mathrm{d}\lambda_{2}\rightarrow
0$

for $n\rightarrow\infty$, which completes the proof. ∎

Altogether we have already established the equivalence of the first five
assertions in Theorem 4.1 and it remains to show the equivalence of the sixth
assertion, which is tackled in the following lemma:

######  Lemma 4.5.

Suppose that $C,C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean copulas
with generators $\psi,\psi_{1},\psi_{2},\ldots$, respectively. Then pointwise
convergence of $(C_{n})_{n\in\mathbb{N}}$ to $C$ is equivalent to pointwise
convergence of $\left(c_{n}^{1:m}\right)_{n\in\mathbb{N}}$ to $c^{1:m}$ almost
everywhere in $\mathbb{I}^{m}$ for some $m\in\\{2,3,\ldots,d-1\\}$.

###### Proof.

We only prove the equivalence for $m=d-1$ since considering the fact that
Cont$(\psi^{(m)})=(0,\infty)$ holds for $m=1,2,\ldots,d-2$ all other cases
follow analogously.
First observe that the set

$\displaystyle\Gamma:=\left\\{\mathbf{x}\in(0,1)^{d-1}:\sum_{i=1}^{d-1}\varphi(x_{i})\in\text{Cont}(D^{-}\psi^{(d-2)})\right\\}\in\mathcal{B}(\mathbb{I}^{d-1})$

has full Lebesgue measure in $\mathbb{I}^{d-1}$. In fact, considering
$\Gamma^{c}:=(0,1)^{d-1}\setminus\Gamma$, applying disintegration and writing
$\mathbf{x}_{1:d-2}=(x_{1},x_{2},\ldots,x_{d-2})$ yields

(4.1)
$\displaystyle\lambda_{d-1}(\Gamma^{c})=\int\limits_{(0,1)^{d-2}}\lambda((\Gamma^{c})_{\mathbf{x}_{1:d-2}})\
\mathrm{d}\lambda_{d-2}(\mathbf{x}_{d-2}).$

For arbitrary $\mathbf{x}_{1:d-2}\in(0,1)^{d-2}$ obviously the
$\mathbf{x}_{1:d-2}$-cut $(\Gamma^{c})_{\mathbf{x}_{1:d-2}}$ of $\Gamma^{c}$
fulfills

$\displaystyle(\Gamma^{c})_{\mathbf{x}_{1:d-2}}=\left\\{x_{d-1}\in(0,1):\sum_{i=1}^{d-1}\varphi(x_{i})\not\in\text{Cont}(D^{-}\psi^{(d-2)})\right\\}.$

Considering the fact that $D^{-}\psi^{(d-2)}$ has at most countably infinitely
many discontinuities and that $\varphi$ is strictly decreasing it follows that
$(\Gamma^{c})_{\mathbf{x}_{1:d-2}}$ is at most countably infinite and
therefore has $\lambda$-measure $0$. Applying eq. (4.1) therefore directly
yields $\lambda_{d-1}(\Gamma^{c})=0$, implying $\lambda_{d-1}(\Gamma)=1$.
In case $\lim_{n\rightarrow\infty}d_{\infty}(C_{n},C)=0$ holds, for
$\mathbf{x}\in\Gamma$ applying Lemma 4.4 yields

$\displaystyle\lim_{n\to\infty}c^{1:d-1}_{n}(\mathbf{x})$
$\displaystyle=\lim_{n\to\infty}\prod_{i=1}^{d-1}\varphi_{n}^{\prime}(x_{i})\cdot
D^{-}\psi_{n}^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi_{n}(x_{i})\right)$
$\displaystyle=\prod_{i=1}^{d-1}\varphi^{\prime}(x_{i})\cdot
D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(x_{i})\right)=c^{1:d-1}(\mathbf{x}).$

Conversely, almost everywhere convergence of $(c^{1:m}_{n})_{n\in\mathbb{N}}$
to $c^{1:m}$ implies almost everywhere convergence of
$(c^{1:2}_{n})_{n\in\mathbb{N}}$ to $c^{1:2}$. Using Scheffé’s theorem we get
that $(C^{1:2}_{n})_{n\in\mathbb{N}}$ converges pointwise to $C^{1:2}$. Since
convergence of bivariate Archimedean copulas is equivalent to pointwise
convergence of $(\varphi_{n})_{n\in\mathbb{N}}$ to $\varphi$ applying Lemma
4.3 yields uniform convergence of $(\psi_{n})_{n\in\mathbb{N}}$ to $\psi$ and
the assertion follows from the already established equivalence of the first
four assertions of Theorem 4.1. ∎

Having proved the equivalence of the six conditions in Theorem 4.1 we now show
that any of these properties implies weak conditional convergence. Notice that
for the case that the (Archimedean) limit copula $C$ is non-strict for every
$z>\varphi(0)$ we obviously have

$\displaystyle\lim_{n\to\infty}(-1)^{d-2}D^{-}\psi^{(d-2)}_{n}(z)=0.$

We now focus on ‘good’ $\mathbf{x}\in\mathbb{I}^{d-1}$ and $y\geq
f^{0}(\mathbf{x})$ and show convergence of the Markov kernels. Notice that the
subsequent lemma already establishes weak conditional convergence for the case
of a strict $d$-dimensional limit copula $C$.

######  Lemma 4.6.

Suppose that $C,C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean copulas
with generators $\psi,\psi_{1},\psi_{2},\ldots$ and corresponding Markov
kernels $K_{C},K_{C_{1}},K_{C_{2}},\ldots$, respectively. If
$(C_{n})_{n\in\mathbb{N}}$ converges uniformly to $C$ then there exists a set
$\Lambda\in\mathcal{B}(\mathbb{I}^{d-1})$ with $\mu_{C^{1:d-1}}(\Lambda)=1$
such that for every $\mathbf{x}\in\Lambda$ the following assertion holds:
there is some set $U^{\mathbf{x}}\subseteq[f^{0}(\mathbf{x}),1]$ which is
dense in $[f^{0}(\mathbf{x}),1]$ and which fulfills that for every $y\in
U^{\mathbf{x}}$

$\displaystyle\lim_{n\to\infty}K_{C_{n}}(\mathbf{x},[0,y])=K_{C}(\mathbf{x},[0,y])$

holds.

###### Proof.

As shown in Lemma 4.5 the set

$\displaystyle\Gamma=\left\\{\mathbf{x}\in(0,1)^{d-1}:\sum_{i=1}^{d-1}\varphi(x_{i})\in\text{Cont}(D^{-}\psi^{(d-2)})\right\\}$

satisfies $\lambda_{d-1}(\Gamma)=1$. Considering
$\mu_{C^{1:d-1}}(\Gamma)=\int_{\Gamma}c^{1:d-1}\ \mathrm{d}\lambda_{d-1}=1$ it
follows that $\Lambda:=\Gamma\setminus L_{0}^{1:d-1}$ fulfills
$\mu_{C^{1:d-1}}(\Lambda)$=1. For fixed $\mathbf{x}\in\Lambda$ it follows by
the same reasoning as in 4.5 that

$\displaystyle
U^{\mathbf{x}}:=\left\\{y\in[f^{0}(\mathbf{x}),1]:\sum_{i=1}^{d-1}\varphi(x_{i})+\varphi(y)\in\text{Cont}(D^{-}\psi^{(d-2)})\right\\}$

is of full $\lambda$-measure in $[f^{0}(\mathbf{x}),1]$. Therefore applying
Lemma 4.4 directly yields

$\displaystyle\lim_{n\to\infty}K_{C_{n}}(\mathbf{x},[0,y])$
$\displaystyle=\lim_{n\to\infty}\frac{D^{-}\psi^{(d-2)}_{n}\left(\sum_{i=1}^{d-1}\varphi_{n}(x_{i})+\varphi_{n}(y)\right)}{D^{-}\psi^{(d-2)}_{n}\left(\sum_{i=1}^{d-1}\varphi_{n}(x_{i})\right)}$
$\displaystyle=\frac{D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(x_{i})+\varphi(y)\right)}{D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(x_{i})\right)}=K_{C}(\mathbf{x},[0,y]).$

∎

It remains to show that we also have convergence for ‘good’ $\mathbf{x}$ and
$y<f^{0}(\mathbf{x})$ (only relevant in the case of non-strict limit $C$.)

######  Lemma 4.7.

Suppose that $C,C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean copulas
with generators $\psi,\psi_{1},\psi_{2},\ldots$ and corresponding Markov
kernels $K_{C},K_{C_{1}},K_{C_{2}},\ldots$, respectively. If
$(C_{n})_{n\in\mathbb{N}}$ converges uniformly to $C$ then there exists a set
$\Lambda\in\mathcal{B}(\mathbb{I}^{d-1})$ with $\mu_{C^{1:d-1}}(\Lambda)=1$
such that for all $\mathbf{x}\in\Lambda$ we have

$\displaystyle\lim_{n\to\infty}K_{C_{n}}(\mathbf{x},[0,y])=0=K_{C}(\mathbf{x},[0,y])$

for every $y\in[0,f^{0}(\mathbf{x}))$.

###### Proof.

Obviously it suffices to consider non-strict $C$. Let $\Lambda$ be as in the
proof of Lemma 4.6 and fix $\mathbf{x}\in\Lambda$ and
$y\in(0,f^{0}(\mathbf{x}))$. Then
$\sum_{i=1}^{d-1}\varphi(x_{i})+\varphi(y)>\varphi(0)$ and we have

$\displaystyle\lim_{n\rightarrow\infty}\sum_{i=1}^{d-1}\varphi_{n}(x_{i})+\varphi_{n}(y)=\sum_{i=1}^{d-1}\varphi(x_{i})+\varphi(y)>\varphi(0).$

Applying Lemma 4.4 (continuous convergence) yields both

$\lim_{n\rightarrow\infty}D^{-}\psi^{(d-2)}_{n}\left(\sum\limits_{i=1}^{d-1}\varphi_{n}(x_{i})+\varphi_{n}(y)\right)=D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(x_{i})+\varphi(y)\right)=0$

and, again using $\mathbf{x}\in\Lambda$,

$\lim_{n\to\infty}D^{-}\psi_{n}^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi_{n}(x_{i})\right)=D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(x_{i})\right)\neq
0.$

Having this, according to eq. 3.3 the desired identity
$\lim_{n\to\infty}K_{C_{n}}(\mathbf{x},[0,y])=0$ follows. Analogous arguments
(or, simply using the fact that for every $d$-dimensional copula $C$ we have
$K_{C}(\mathbf{x},\\{0\\})=0$ for $\mu_{C^{1:d-1}}$-almost every
$\mathbf{x}\in\mathbb{I}^{d-1}$) also apply in the case $y=0$, so the proof is
complete. ∎

######  Remark 4.8.

The proof of Lemma 4.7 is also applicable in dimension $d=2$ and therefore
provides an alternative simpler version of the rather involved approach
followed in [14].

Theorem 4.1 has the following straightforward consequence (analogous to the
bivariate statement considered Theorem 4.2 in [14]):

######  Theorem 4.9.

Suppose that $C,C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean copulas
with generators $\psi,\psi_{1},\psi_{2},\ldots$, respectively. Then the
following assertions are equivalent:

  1. (1)

$(C_{n})_{n\in\mathbb{N}}$ converges uniformly to $C$.

  2. (2)

$(\varphi_{n})_{n\in\mathbb{N}}$ converges pointwise to $\varphi$ on $(0,1]$.

  3. (3)

$(\psi_{n})_{n\in\mathbb{N}}$ converges uniformly to $\psi$ on $[0,\infty)$.

  4. (4)

$(\psi_{n}^{(m)})_{n\in\mathbb{N}}$ converges pointwise to $\psi^{(m)}$ on
$(0,\infty)$ for every $m\in\\{1,2,\ldots,d-2\\}$ and
$(D^{-}\psi_{n}^{(d-2)})_{n\in\mathbb{N}}$ converges pointwise to
$D^{-}\psi^{(d-2)}$ on _Cont_ $(D^{-}\psi^{(d-2)})$.

  5. (5)

$(C_{n})_{n\in\mathbb{N}}$ converges weakly conditional to $C$ and
$(c_{n}^{1:d-1})_{n\in\mathbb{N}}$ converges to $c^{1:d-1}$
$\lambda_{d}$-almost everywhere in $\mathbb{I}^{d-1}$.

###### Proof.

We already know that the first four conditions are equivalent and that each of
them implies the fifth assertion. On the other hand, if the conditions in (5)
hold, then we also have $\lambda_{m}$-almost everywhere convergence of
$(c_{n}^{1:m})_{n\in\mathbb{N}}$ to $c^{1:m}$ for every
$m\in\\{2,\ldots,d-2\\}$ and applying Theorem 4.1 completes the proof. ∎

##  5\. Archimedean copulas and the Williamson transform

Following [25], every $d$-monotone function can be represented via the so-
called Williamson transform of a unique probability measure on $[0,\infty)$.
As a consequence, we may describe and handle $d$-dimensional Archimedean
copulas $C=C_{\psi}$ in terms of their corresponding probability measure
$\gamma$ on $[0,\infty)$. In what follows we first establish some
complementary useful results describing the interrelation between generator
$\psi$ and probability measures $\gamma$, express masses of level sets as well
as the Kendall distribution function handily in terms of $\gamma$, show that
regularity properties of the corresponding measure $\gamma$ carry over to the
Archimedean copula and prove the fact that pointwise convergence of
Archimedean copulas $C_{1},C_{2},\ldots$ to an Archimedean copula $C$ is
equivalent to weak convergence of the corresponding probability measures
$\gamma_{1},\gamma_{2},\ldots$ to the probability measure $\gamma$. Based on
these facts we then finally show that both, the family of absolutely
continuous and the family of singular Archimedian copulas is dense in
$(\mathcal{C}_{ar}^{d},d_{\infty})$.

We start with recalling the following result (see [21] and [25]) where we
write $f_{+}^{m}$ for the $m$-th power of the positive part $f_{+}$ of a
function $f$, i.e., $f_{+}^{m}:=(f_{+})^{m}$:

######  Theorem 5.1.

Let $\psi\colon[0,\infty)\rightarrow\mathbb{I}$ be a function and $d\geq 2$.
Then the following two conditions are equivalent:

  * (1)

$\psi$ is the generator of a $d$-dimensional Archimedean copula $C_{\psi}$.

  * (2)

There exists a unique probability measure $\gamma$ on
$\mathcal{B}([0,\infty))$ with $\gamma(\\{0\\})=0$ such that

(5.1)
$\psi(z)=\int_{[0,\infty)}(1-tz)_{+}^{d-1}\mathrm{d}\gamma(t)=:(\mathcal{W}_{d}\gamma)(z),$

holds for every $z>0$. In other words, $\psi$ is the Williamson transform
$\mathcal{W}_{d}\gamma$ of $\gamma$.

Obviously our assumed normalization property $\psi(1)=\frac{1}{2}$ translates
to

(5.2) $\int_{\mathbb{I}}(1-t)^{d-1}\mathrm{d}\gamma(t)=\frac{1}{2}.$

In what follows we therefore only consider measures $\gamma$ fulfilling eq.
(5.2), let $\mathcal{P}_{\mathcal{W}}$ denote the family of all these
measures, i.e.,

(5.3)
$\mathcal{P}_{\mathcal{W}}=\left\\{\gamma\in\mathcal{P}([0,\infty)):\gamma(\\{0\\})=0\textrm{
and }\int_{\mathbb{I}}(1-t)^{d-1}\mathrm{d}\gamma(t)=\frac{1}{2}\right\\}$

and refer to $\mathcal{P}_{\mathcal{W}_{d}}$ as the family of all
$d$-Williamson measures.

Again following [21] next we derive an explicit representation for the
cumulative distribution function of $\gamma$ in terms of the Archimedean
generator $\psi$:

######  Lemma 5.2.

Let $\psi$ be the generator of a $d$-dimensional Archimedean copula and
$\gamma\in\mathcal{P}_{\mathcal{W}_{d}}$ its corresponding Williamson measure.
Then

(5.4)
$\gamma([0,z])=\sum_{k=0}^{d-2}\frac{(-1)^{k}\psi^{(k)}(\tfrac{1}{z})}{k!}\frac{1}{z^{k}}+\frac{(-1)^{d-1}D^{-}\psi^{(d-2)}(\tfrac{1}{z})}{(d-1)!}\frac{1}{z^{d-1}}$

holds for every $z>0$.

###### Proof.

Defining $F\colon(0,\infty)\rightarrow\mathbb{I}$ by
$F(z):=\gamma([0,z])=\gamma((0,z])$ according to [27]

(5.5)
$F(z)=\sum_{k=0}^{d-1}\frac{(-1)^{k}\psi^{(k)}(\tfrac{1}{z})}{k!}\frac{1}{z^{k}}$

holds for every continuity point $z$ of $F$, i.e., for every continuity point
$z$ of $D^{-}\psi^{(d-2)}$. Set $\text{Cont}(D^{-}\psi^{(d-2)})$ and define
the function $G:(0,\infty)\rightarrow\mathbb{I}$ by

$\displaystyle
G(z):=\sum_{k=0}^{d-2}\frac{(-1)^{k}\psi^{(k)}(\frac{1}{z})}{k!}\frac{1}{z^{k}}+\frac{(-1)^{(d-1)}D^{-}\psi^{(d-2)}(\frac{1}{z})}{(d-1)!}\frac{1}{z^{d-1}}.$

Then the distribution function $F$ and the right-continuous function $G$
coincide on $\text{Cont}(D^{-}\psi^{(d-2)})$ and since the latter is dense in
$(0,\infty)$ equality $F=G$ follows and the proof is complete. ∎

The following lemma expresses $D^{-}\psi^{(d-2)}$ (appearing both in the
numerator and the denominator of the Markov kernel $K_{C}(\cdot,\cdot)$ in
Theorem 3.1) in terms of $\gamma$ and will be useful in the sequel:

######  Lemma 5.3.

Let $\psi$ be the generator of a $d$-dimensional Archimedean copula and
$\gamma\in\mathcal{P}_{\mathcal{W}_{d}}$ be the corresponding Williamson
measure. Then

$0\geq
G(z):=(-1)^{d-2}D^{-}\psi^{(d-2)}(z)=-(d-1)!\int_{(0,\frac{1}{z}]}t^{d-1}\mathrm{d}\gamma(t)$

holds for every $z>0$.

###### Proof.

According to [25] we already know that for every $z>0$ we have

$(-1)^{d-2}\psi^{d-2}(z)=(d-1)!\int_{[0,\infty)}t^{d-2}(1-zt)_{+}\mathrm{d}\gamma(t)=(d-1)!\int_{(0,\infty)}t^{d-2}(1-zt)_{+}\mathrm{d}\gamma(t).$

Since $(-1)^{d-2}\psi^{d-2}$ is convex, the left hand derivative exists
everywhere in $(0,\infty)$ and is left-continuous. For fixed $z>0$ considering
the left-hand difference quotient

$Q_{h}(z):=(d-1)!\,\int_{(0,\infty)}t^{d-2}\frac{(1-t(z+h))_{+}-(1-tz)_{+}}{h}\mathrm{d}\gamma(t)$

for $h\in(-z,0)$ we get

$\displaystyle Q_{h}(z)$
$\displaystyle=\frac{(d-1)!}{h}\int_{(\frac{1}{z},\frac{1}{z+h})}t^{d-2}(1-t(z+h))\mathrm{d}\gamma(t)$
$\displaystyle\qquad+\frac{(d-1)!}{h}\int_{(0,\frac{1}{z}]}t^{d-2}[(1-t(z+h))-(1-tz)]\mathrm{d}\gamma(t)$
$\displaystyle=\underbrace{\frac{(d-1)!}{h}\int_{(\frac{1}{z},\frac{1}{z+h})}t^{d-2}(1-t(z+h))\mathrm{d}\gamma(t)}_{=:I_{h}}\,\,-(d-1)!\,\int_{[0,\frac{1}{z}]}t^{d-1}\mathrm{d}\gamma(t),$

and it suffices to show that $I_{h}$ converges to $0$. Using the monotonicity
and non-negativity of the functions $t\mapsto t^{d-2}$ and $t\mapsto 1-t(z+h)$
on $(\frac{1}{z},\frac{1}{z+h})$ it follows that

$\displaystyle|I_{h}|$
$\displaystyle\leq\frac{1}{|h|}\frac{1}{(z+h)^{d-2}}\left(1-\frac{z+h}{z}\right)\gamma\left(\left(\frac{1}{z},\frac{1}{z+h}\right)\right)$
$\displaystyle=\frac{1}{(z+h)^{d-2}\,z}\,\,\gamma\left(\left(\frac{1}{z},\frac{1}{z+h}\right)\right)$

from which the assertion follows immediately. ∎

As first application of the previous lemma we characterize strictness in terms
of the Williamson measure $\gamma$:

######  Lemma 5.4.

Suppose that $C$ is an Archimedean copula with Williamson measure
$\gamma\in\mathcal{P}_{\mathcal{W}_{d}}$. Then $C$ is strict if, and only if
the support of $\gamma$ contains $0$, i.e., if $\gamma([0,r))>0$ for every
$r>0$.

###### Proof.

If the support of $\gamma$ contains $0$ then obviously
$\int_{(0,r)}t^{d-1}\mathrm{d}\gamma(t)>0$ for every $r>0$, hence applying
Lemma 5.3 directly yields $(-1)^{d-1}D^{-}\psi^{(d-2)}(z)>0$ for every $z>0$.
Having this, strictness of $\psi$ follows immediately.
On the other hand, if there exists some $r>0$ with $\gamma([0,r))=0$ then
considering $z_{0}=\frac{1}{2r}$ and again using Lemma 5.3 we have
$(-1)^{d-1}D^{-}\psi^{(d-2)}(z_{0})=0$. Since the function
$z\mapsto(-1)^{d-1}D^{-}\psi^{(d-2)}(z)$ is non-negative, non-increasing and
left-continuous it follows that $(-1)^{d-1}D^{-}\psi^{(d-2)}(z)=0$ holds for
every $z\geq z_{0}$. Hence $\psi(z_{0})=0$ and $\psi$ is non-strict. ∎

We now return to the formulas for the level set masses and the Kendall
distribution function of Archimedean copulas as already mentioned in Section 3
and reformulate them elegantly in terms of the Williamson measure $\gamma$.
Although surprising, to the best of the authors’ knowledge these formulas seem
to be new:

######  Theorem 5.5.

Let $C$ be a $d$-dimensional Archimedean copula with generator $\psi$ and
Williamson measure $\gamma$. Then (compare with equations (3.5) - (3.7)):

(5.6)
$\displaystyle\mu_{C}(L_{t})=\gamma(\\{\tfrac{1}{\varphi(t)}\\}),\,t\in(0,1]$

holds for every $t\in(0,1]$ and every $C$. Furthermore, for strict $C$ we have
$\mu_{C}(L_{0})=0$, and for non-strict $C$

(5.7) $\displaystyle\mu_{C}(L_{0})=\gamma(\\{\tfrac{1}{\varphi(0)}\\}).$

holds. Finally, the Kendall distribution function $F_{K}^{d}$ of $C$ fulfills

(5.8) $\displaystyle F_{K}^{d}(t)=\gamma([0,\tfrac{1}{\varphi(t)}])$

for every $t\in(0,1]$.

###### Proof.

First of all notice that the expression for the Kendall distribution function
follows immediately from equation (5.4). Furthermore for $t\in(0,1]$
considering

$\mu_{C}(L_{t})=F_{K}^{d}(t)-F_{K}^{d}(t-)$

eq. (5.6) follows immediately from eq. (5.8). Finally, using eq. (3.6) and
incorporating Lemma 5.3 yields

$\displaystyle\mu_{C}(L_{0})=\frac{(-1)^{d-1}(\varphi(0))^{d-1}}{(d-1)!}D^{-}\psi^{(d-2)}(\varphi(0))=\varphi(0)^{d-1}\int_{(0,\frac{1}{\varphi(0)}]}t^{d-1}\mathrm{d}\gamma(t).$

Since eq. (5.1) implies that for every $z_{0}>0$ we have that $\psi(z_{0})=0$
is equivalent to $\gamma((0,\tfrac{1}{z_{0}}))=0$, the right-hand side of the
last equation simplifies to $\gamma(\\{\tfrac{1}{\varphi(0)}\\})$ and the
proof is complete. ∎

######  Remark 5.6.

Notice that eq. (5.8) implies $F_{K}^{d}(\frac{1}{2})=\gamma([0,1])$. More
importantly, the (probably most) famous conjecture in the context of copulas,
saying that for every fixed $d\geq 3$ two Archimedean copulas
$C,D\in\mathcal{C}^{d}_{ar}$ are identical if, and only if their Kendall
distribution functions coincide (see [7] and [8]) would follow if it could be
shown that the mapping assigning each Williamson measure $\gamma$ the function
$F_{\gamma}:\mathbb{I}\rightarrow\mathbb{I}$, defined by

$F_{\gamma}(t)=\gamma\left(\left[0,\frac{1}{\varphi_{\gamma}(t)}\right]\right)$

is injective, where $\varphi_{\gamma}$ denotes the pseudo-inverse of the
generator $\psi=\mathcal{W}_{d}\,\gamma$.

######  Example 5.7.

The probability measure
$\gamma=\frac{7}{8}\,\delta_{1/4}+\frac{1}{8}\,\delta_{3/4}$ obviously
fulfills $\gamma\in\mathcal{P}_{\mathcal{W}_{3}}$. The induced generator
$\psi$ is given by

$\displaystyle\psi(z):=\begin{cases}1-\frac{5z}{8}+\frac{z^{2}}{8}&\text{ if
}z<\frac{4}{3}\\\ \frac{7}{8}\left(1-\frac{z}{4}\right)^{2}&\text{ if
}z\in\left[\frac{4}{3},4\right]\\\ 0&\text{ otherwise}\end{cases}$

and it is straightforward to verify that $\varphi(\frac{7}{18})=\frac{4}{3}$
and $\varphi(0)=4$ holds. Using Theorem 5.5 therefore yields
$\mu_{C}(L_{0})=\frac{7}{8}$ as well as $\mu_{C}(L_{7/18})=\frac{1}{8}$.
Figure 2 depicts the distribution function $z\mapsto\gamma([0,z])$ of $\gamma$
(left panel), the induced generator $\psi$ (middle) and the sets $L_{0}$ and
$L_{7/18}$ carrying the mass (right panel).

Figure 2. Distribution function of $\gamma$ (left panel), induced generator
$\psi$ (middle) and the sets $L_{0}$ and $L_{7/18}$ (right panel) as
considered in Example 5.7.

The next result complements Theorem 4.1 and adds a seventh equivalent
condition in terms of the corresponding Williamson measures. During the
process of preparing this manuscript it has been brought to our attention that
this very result was already established in [1]. Considering that the result
is key especially for the subsequent regularity results and that the
subsequent proof is simpler and less technical than the one given in [1] we
include it for the sake of completeness.

######  Theorem 5.8.

Suppose that $C,C_{1},C_{2},\ldots$ are $d$-dimensional Archimedean copulas
with generators $\psi,\psi_{1},\psi_{2},\ldots$ and let
$\gamma,\gamma_{1},\gamma_{2},\ldots$ denote the corresponding Williamson
measures. Then the following assertions are equivalent:

  * (1)

$(C_{n})_{n\in\mathbb{N}}$ converges uniformly to $C$.

  * (2)

$(\gamma_{n})_{n\in\mathbb{N}}$ converges weakly on $[0,\infty)$ to $\gamma$.

###### Proof.

According to Theorem 4.1 the first assertion is equivalent to uniform
convergence of $(\psi_{n})_{n\in\mathbb{N}}$ to $\psi$. (i) If
$(\gamma_{n})_{n\in\mathbb{N}}$ converges weakly to $\gamma$, then applying
Theorem 5.1 and using the fact that the function $t\mapsto(1-tz)_{+}^{d-1}$ is
continuous and bounded

$\psi_{n}(z)=\int_{[0,\infty)}(1-tz)_{+}^{d-1}d\gamma_{n}(t)\overset{n\rightarrow\infty}{\longrightarrow}\int_{[0,\infty)}(1-tz)_{+}^{d-1}d\gamma(t)=\psi(z)$

follows for every fixed but arbitrary $z\in[0,\infty)$. (ii) Vice versa, using
Lemma 5.2 and Lemma 4.4 and considering $z\in(0,\infty)$ with
$\frac{1}{z}\in\text{Cont}(D^{-}\psi^{(d-2)})$ yields

$\displaystyle\lim_{n\rightarrow\infty}\gamma_{n}([0,z])$
$\displaystyle=\lim_{n\rightarrow\infty}\sum_{k=0}^{d-2}\frac{(-1)^{k}\psi_{n}^{(k)}(\tfrac{1}{z})}{k!}z^{-k}+\frac{(-1)^{d-1}D^{-}\psi_{n}^{(d-2)}(\tfrac{1}{z})}{(d-1)!}z^{-d+1}$
$\displaystyle=\sum_{k=0}^{d-2}\frac{(-1)^{k}\psi^{(k)}(\tfrac{1}{z})}{k!}z^{-k}+\frac{(-1)^{d-1}D^{-}\psi^{(d-2)}(\tfrac{1}{z})}{(d-1)!}z^{-d+1}$
$\displaystyle=\gamma([0,z]).$

This completes the proof since, firstly,
$(0,1)\setminus\text{Cont}(D^{-}\psi^{(d-2)})$ is at most countably infinite
and, secondly, convergence of distribution functions on a dense set implies
weak convergence. ∎

We now focus on studying how regularity/singularity properties of the
Williamson measure carries over to regularity/singularity properties of the
corresponding copula $C_{\gamma}\in\mathcal{C}^{d}_{ar}$ and first recall some
basic notation. For every $m\in\\{1,\ldots,d\\}$ we say that a finite measure
$\vartheta$ on $\mathcal{B}(\mathbb{I}^{m})$ is singular (with respect to
$\lambda_{m}$) if there exists some $G\in\mathcal{B}(\mathbb{I}^{m})$
fulfilling $\vartheta(G)=\vartheta(\mathbb{I}^{m})$ and $\lambda_{d}(G)=0$. A
copula $C\in\mathcal{C}^{m}$ is called singular if the corresponding
$m$-stochastic measure $\mu_{C}$ is singular.

For the bivariate setting singularity of a copula $C$ is equivalent to
singularity of $\lambda$-almost all conditional distributions $K_{C}(x,\cdot)$
(see Lemma 1 in [5]). A a fully analogous statement can not hold in general
for arbitrary $d\geq 3$ \- in fact, for example the copula $C$ of a random
vector $(X,X,Y)$ with $X,Y$ being independent and uniformly distributed on
$[0,1]$ is obviously singular but $\mu_{C^{1:2}}=\mu_{M}$-almost every
conditional distribution $K_{C}(x,x,\cdot)$ coincides with $\lambda$ and
therefore is absolutely continuous. Assuming, however, absolute continuity of
$C^{1:d-1}$ as it is the case for every $C\in\mathcal{C}^{d}_{ar}$, an
analogue of the bivariate result remains valid:

######  Lemma 5.9.

Let $C$ be a $d$-dimensional copula such that $C^{1:d-1}$ is absolutely
continuous. Then $C$ is singular if, and only if there exists some set
$\Lambda\in\mathcal{B}(\mathbb{I}^{d-1})$ fulfilling
$\mu_{C^{1:d-1}}(\Lambda)=1$ such that $K_{C}(\mathbf{x},\cdot)$ is singular
for every $\mathbf{x}\in\Lambda$.

###### Proof.

If $C\in\mathcal{C}^{d}$ is singular then by definition there exists some set
$G\in\mathcal{B}(\mathbb{I}^{d-1})$ fulfilling $\mu_{C}(G)=1$ as well as
$\lambda_{d}(G)=0$. Since every set $G\in\mathcal{B}(\mathbb{I}^{d-1})$ with
$\lambda_{d-1}(G)=1$ also fulfills $\mu_{C^{1:d-1}}(G)=1$, disintegration
implies the existence of a set $\Lambda\in\mathcal{B}(\mathbb{I}^{d-1})$ with
$\lambda_{d-1}(\Lambda)=1$ such that for every $\mathbf{x}\in\Lambda$ we have
that the $\mathbf{x}$-cut $G_{\mathbf{x}}$ of $G$ fulfills
$K_{C}(\mathbf{x},G_{\mathbf{x}})=1$ and $\lambda(G_{\mathbf{x}})=0$. In other
words, $K_{C}(\mathbf{x},\cdot)$ is singular and the first implication is
proved.

The reverse implication can be proved as follows. We show the contraposition
and assume that $\mu_{C}$ is not singular with respect to $\lambda_{d}$, i.e.,
the absolutely continuous part $\mu_{C}^{abs}$ of the Lebesgue decomposition
$\mu_{C}=\mu_{C}^{abs}+\mu_{C}^{sing}$ of $\mu_{C}$ with respect to
$\lambda_{d}$ is non-degenerated in the sense that
$\mu_{C}^{abs}(\mathbb{I}^{d})>0$ holds. Let
$G\in\mathcal{B}(\mathbb{I}^{d-1})$ with $\lambda_{d-1}(G)=0$ be arbitrary but
fixed. Then obviously

$(\mu_{C}^{sing})^{1:d-1}(G)=\mu_{C}^{sing}(G\times\mathbb{I})\leq\mu_{C}(G\times\mathbb{I})=\mu_{C}^{1:d-1}(G)=\int_{G}c^{1:d-1}(\mathbf{x})\mathrm{d}\lambda_{d-1}=0,$

so there exists a Radon-Nikodym derivative
$f\colon\mathbb{I}^{d-1}\rightarrow[0,\infty)$ of $(\mu_{C}^{sing})^{1:d-1}$
with respect to $\lambda_{d-1}$. Letting
$k\colon\mathbb{I}^{d}\rightarrow[0,\infty)$ denote the Radon-Nikodym
derivative of $\mu_{C}^{abs}$ with respect to $\lambda_{d}$, using
disintegration, for arbitrary $E\in\mathcal{B}(\mathbb{I}^{d-1})$,
$F\in\mathcal{B}(\mathbb{I})$ we get

$\displaystyle\int_{E}K_{C}(\mathbf{x},F)c^{1:d-1}$
$\displaystyle(\mathbf{x})\mathrm{d}\lambda_{d-1}(\mathbf{x})=\int_{E}K_{C}(\mathbf{x},F)\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{x})$
$\displaystyle=\mu_{C}(E\times F)=\mu_{C}^{abs}(E\times
F)+\mu_{C}^{sing}(E\times F)$
$\displaystyle=\int_{E}\left[\int_{F}k(\mathbf{x},y)\mathrm{d}\lambda(y)\right]\mathrm{d}\lambda_{d-1}(\mathbf{x})+\int_{E}H^{sing}(\mathbf{x},F)\mathrm{d}(\mu_{C}^{sing})^{1:d-1}(\mathbf{x})$
$\displaystyle=\int_{E}\left[\int_{F}k(\mathbf{x},y)\mathrm{d}\lambda(y)\right]\mathrm{d}\lambda_{d-1}(\mathbf{x})+\int_{E}H^{sing}(\mathbf{x},F)f(\mathbf{x})\mathrm{d}\lambda_{d-1}(\mathbf{x})$
$\displaystyle=\int_{E}\left[\int_{F}k(\mathbf{x},y)\mathrm{d}\lambda(y)+H^{sing}(\mathbf{x},F)f(\mathbf{x})\right]\mathrm{d}\lambda_{d-1}(\mathbf{x})$

where $H^{sing}(\mathbf{x},\cdot)$ denotes the conditional measure (sub- or
super Markov kernel) of $\mu_{C}^{sing}$ given $\mathbf{x}$. Since
$E\in\mathcal{B}(\mathbb{I}^{d-1})$ it follows that

$K_{C}(\mathbf{x},F)c^{1:d-1}(\mathbf{x})=\int_{F}k(\mathbf{x},y)\mathrm{d}\lambda(y)+H^{sing}(\mathbf{x},F)f(\mathbf{x})$

holds for $\lambda_{d-1}$-almost every $\mathbf{x}\in\mathbb{I}^{d-1}$. Using
the fact that $\mu_{C}^{1:d-1}$ is absolutely continuous and that obviously
$\mu_{C^{1:d-1}}(\\{\mathbf{x}\in\mathbb{I}^{d-1}\colon
c^{1:d-1}(\mathbf{x})=0\\})=0$ yields the identity

$K_{C}(\mathbf{x},F)=\int_{F}\frac{k(\mathbf{x},y)}{c^{1:d-1}(\mathbf{x})}\mathrm{d}\lambda(y)+H^{sing}(\mathbf{x},F)\frac{f(\mathbf{x})}{c^{1:d-1}(\mathbf{x})}$

for $\mu_{C^{1:d-1}}$-almost every $\mathbf{x}\in\mathbb{I}^{d-1}$. Since
$\mu_{C}^{abs}$ is non-degenerated by assumption, there exists a set
$\Upsilon\in\mathcal{B}(\mathbb{I}^{d-1})$ with $c^{1:d-1}(\mathbf{x})>0$ for
every $\mathbf{x}\in\Upsilon$ and $\mu_{C^{1:d-1}}(\Upsilon)>0$ such that for
every $\mathbf{x}\in\Upsilon$ the absolutely continuous measure
$F\mapsto\int_{F}\frac{k(\mathbf{x},y)}{c^{1:d-1}(\mathbf{x})}\mathrm{d}\lambda(y)$
is non-degenerated. This shows that for such $\mathbf{x}$ the measure
$K_{C}(\mathbf{x},\cdot)$ can not be singular and the proof is complete. ∎

Following [20] every Markov kernel
$K_{C}(\cdot,\cdot)\colon\mathbb{I}^{d-1}\times\mathcal{B}(\mathbb{I})\rightarrow\mathbb{I}$
can be decomposed into the sum of three sub- Markov kernels from $\mathbb{I}$
to $\mathcal{B}(\mathbb{I})$ as

(5.9) $\displaystyle
K_{C}(\mathbf{x},\cdot)=K_{C}^{dis}(\mathbf{x},\cdot)+K_{C}^{sing}(\mathbf{x},\cdot)+K_{C}^{abs}(\mathbf{x},\cdot),$

whereby each measure $K_{C}^{dis}(\mathbf{x},\cdot)$ is discrete, each
$K_{C}^{sing}(\mathbf{x},\cdot)$ is singular and has no point masses and
$K_{C}^{abs}(\mathbf{x},\cdot)$ is absolutely continuous on
$\mathcal{B}(\mathbb{I})$. Again assuming absolute continuity of $C^{1:d-1}$
and letting $c^{1:d-1}$ denote the corresponding density in what follows we
will refer to the three measures $\mu_{C}^{dis},\mu_{C}^{sing},\mu_{C}^{abs}$,
defined by

$\displaystyle\mu_{C}^{dis}(G)$
$\displaystyle=\int_{\mathbb{I}^{d-1}}K_{C}^{dis}(\mathbf{x},G_{\mathbf{x}})c^{1:d-1}(\mathbf{x})\mathrm{d}\lambda_{d-1}(\mathbf{x})$
(5.10) $\displaystyle\mu_{C}^{sing}(G)$
$\displaystyle=\int_{\mathbb{I}^{d-1}}K_{C}^{sing}(\mathbf{x},G_{\mathbf{x}})c^{1:d-1}(\mathbf{x})\mathrm{d}\lambda_{d-1}(\mathbf{x})$
$\displaystyle\mu_{C}^{abs}(G)$
$\displaystyle=\int_{\mathbb{I}^{d-1}}K_{C}^{abs}(\mathbf{x},G_{\mathbf{x}})c^{1:d-1}(\mathbf{x})\mathrm{d}\lambda_{d-1}(\mathbf{x})$

for every $G\in\mathcal{B}(\mathbb{I})$ as the discrete, the singular, and the
absolutely continuous component of $\mu_{C}$.

We now show how singularity/regularity of $\gamma$ carries over to the
corresponding Archimedean copula.

######  Theorem 5.10.

Suppose that $C\in\mathcal{C}_{ar}^{d}$ has generator $\psi$ and Williamson
measure $\gamma\in\mathcal{P}_{\mathcal{W}_{d}}$. Then the following
assertions hold:

  * (1)

If $\gamma$ is absolutely continuous then $\mu_{C}^{abs}(\mathbb{I}^{d})=1$,
i.e., $C$ is absolutely continuous.

  * (2)

If $\gamma$ is discrete then $\mu_{C}^{dis}(\mathbb{I}^{d})=1$.

  * (3)

If $\gamma$ is singular without point masses then
$\mu_{C}^{sing}(\mathbb{I}^{d})=1$.

###### Proof.

(i) The first assertion has already been established in [21] and can
alternatively be proved easily as follows: Suppose that $\gamma$ is absolutely
continuous with density $f$. Then using Lemma 5.3 we have

$(-1)^{d-2}D^{-}\psi^{(d-2)}(z)=-(d-1)!\int_{(0,\frac{1}{z}]}t^{d-1}f(t)\mathrm{d}\lambda(t).$

Considering that the right-hand side is obviously continuous in $z$ it follows
that $\psi^{(d-1)}$ exists on the full interval $(0,\infty)$. Moreover, the
right-hand side is easily seen to be absolutely continuous, hence Proposition
4.2 in [21] yields absolute continuity of $C$.
For the proof of the remaining two assertions first notice that is suffices to
consider $\mathbf{x}$ fulfilling $M(\mathbf{x})<1$ and $\mathbf{x}\not\in
L_{0}^{1:d-1}$ and that in this case we have
$0<\sum_{i=1}^{d-1}\varphi(x_{i})<\varphi(0)\in(0,\infty]$ as well as
$(-1)^{d-1}D^{-}\psi^{(d-2)}\left(\sum_{i=1}^{d-1}\varphi(x_{i})\right)>0$.
Moreover, additionally assuming $y\geq f^{0}(\mathbf{x})$ using Lemma 5.3 the
Markov kernel $K_{C}(\cdot,\cdot)$ according to eq. (3.3) can be expressed as

(5.11) $\displaystyle K_{C}(\mathbf{x},[0,y])$
$\displaystyle=\frac{\int_{I_{y}}t^{d-1}\mathrm{d}\gamma(t)}{\int_{I_{1}}t^{d-1}\mathrm{d}\gamma(t)}$

with
$I_{y}=\left(0,\frac{1}{\sum_{i=1}^{d-1}\varphi(x_{i})\,+\varphi(y)}\right]$
for every $y\in\mathbb{I}$.
(ii) Suppose now that $\gamma$ is discrete. Then there exist
$a_{1},a_{2},\ldots\in(0,\infty)$ and constants
$\alpha_{1},\alpha_{2},\ldots\in\mathbb{I}$ with
$\sum_{j=1}^{\infty}\alpha_{j}=1$ such that
$\gamma=\sum_{j=1}^{\infty}\alpha_{j}\delta_{a_{j}}$ holds, and eq. (5.11)
simplifies to

(5.12) $\displaystyle K_{C}(\mathbf{x},[0,y])$
$\displaystyle=\frac{\sum_{j:a_{j}\in
I_{y}}a_{j}^{d-1}\alpha_{j}}{\sum_{j:a_{j}\in I_{1}}a_{j}^{d-1}\alpha_{j}}.$

Notice that we do not assume all $\alpha_{j}$ to be greater than zero, so the
case of finitely many point masses is covered as well. Considering, firstly,
that $K_{C}(\mathbf{x},[0,y])=0$ for $y<f^{0}(\mathbf{x})$ and that the
condition $a_{j}\in I_{y}$ is equivalent to
$y\geq\psi(\frac{1}{a_{j}}-\sum_{i=1}^{d-1}\varphi(x_{i}))$ the function
$y\mapsto K_{C}(\mathbf{x},[0,y])$ is easily seen to be the distribution
function of the discrete probability measure on $\mathbb{I}$ having point mass
$\frac{a_{j}^{d-1}\alpha_{j}}{\sum_{l:a_{l}\in I_{1}}a_{l}^{d-1}\alpha_{l}}$
in $\psi(\frac{1}{a_{j}}-\sum_{i=1}^{d-1}\varphi(x_{i}))$ for every $j$ with
$a_{j}\in I_{1}$. In other words, $K_{C}(\mathbf{x},\cdot)$ is a discrete
probability measure, so
$K_{C}(\mathbf{x},\cdot)=K_{C}^{dis}(\mathbf{x},\cdot)$ holds for all
$\mathbf{x}$ fulfilling $M(\mathbf{x})<1$ and $\mathbf{x}\not\in
L_{0}^{1:d-1}$. Having this and using eq. (5)
$\mu_{C}^{dis}(\mathbb{I}^{d})=1$ follows.
(iii) Finally suppose that $\gamma$ is singular without point masses and again
consider some $\mathbf{x}$ fulfilling $M(\mathbf{x})<1$ and $\mathbf{x}\not\in
L_{0}^{1:d-1}$. Then Theorem 5.5 implies that we have $\mu_{C}(L_{t})=0$ for
every $t\in\mathbb{I}$ so the conditional distribution function $y\mapsto
F^{C}_{\mathbf{x}}(y)=K_{C}(\mathbf{x},[0,y])$ is continuous and it suffices
to show that its derivative $(F^{C}_{\mathbf{x}})^{\prime}$ fulfills
$(F^{C}_{\mathbf{x}})^{\prime}(y)=0$ for $\lambda$-almost every
$y>f^{0}(\mathbf{x})$, which can be done as follows: As already mentioned
before, our choice of $\mathbf{x}$ implies that
$0<\sum_{i=1}^{d-1}\varphi(x_{i})<\varphi(0)\in(0,\infty]$, so in particular

$\frac{1}{\sum_{i=1}^{d-1}\varphi(x_{i})}>\frac{1}{\varphi(0)}$

and therefore $\gamma(I_{1})>0$. Defining the measure
$m:\mathcal{B}((0,\infty))\rightarrow[0,\infty]$ by

$m(B):=\int_{B}t^{d-1}\mathrm{d}\gamma(t)$

it follows that $m$ is $\sigma$-finite (in fact, finite for every finite
interval), singular with respect to $\lambda$, has no point masses and
fulfills $0<m(I_{1})<\infty$. Letting $G_{m}:I_{1}\rightarrow[0,\infty)$
denote the measure-generating function induced by $m$ via $G_{m}(x):=m([0,x])$
singularity of $m$ implies that the set $\Lambda$, defined by

$\Lambda:=\\{z\in I_{1}:G_{m}^{\prime}(z)=0\\}$

fulfills $\lambda(\Lambda)=\lambda(I_{1})$. Defining $\Upsilon$ by

$\Upsilon:=\left\\{y\in(f^{0}(\mathbf{x}),1]:\,\frac{1}{\sum_{i=1}^{d-1}\varphi(x_{i})+\varphi(y)}\in\Lambda\right\\}\in\mathcal{B}(\mathbb{I}),$

using the fact that $\varphi$ is differentiable and strictly decreasing on
$(0,1)$ with derivative bounded away from $0$ on any compact interval
$[a,b]\subseteq(0,1)$ it follows (see [13]) that
$\lambda(\Upsilon)=\lambda((f^{0}(\mathbf{x}),1])$. For every $y\in\Upsilon$,
however, the chain rule yields

$(F^{C}_{\mathbf{x}})^{\prime}(y)=\frac{1}{m(I_{1})}\,\underbrace{G_{m}^{\prime}\left(\frac{1}{\sum_{i=1}^{d-1}\varphi(x_{i})+\varphi(y)}\right)}_{=0}\cdot\frac{\partial}{\partial
y}\left(\frac{1}{\sum_{i=1}^{d-1}\varphi(x_{i})+\varphi(y)}\right)=0.$

Altogether we have shown that for arbitrary $\mathbf{x}$ fulfilling
$M(\mathbf{x})<1$ and $\mathbf{x}\not\in L_{0}^{1:d-1}$ the measure
$K_{C}(\mathbf{x},\cdot)$ is singular without point masses, i.e.,
$K_{C}(\mathbf{x},\cdot)=K_{C}^{sing}(\mathbf{x},\cdot)$ holds and considering
eq. (5) again $\mu_{C}^{sing}(\mathbb{I}^{d})=1$ follows. ∎

######  Remark 5.11.

The second assertion of Theorem 5.10 can be proved in the following
alternative way (the afore-mentioned version was chosen in order to underline
the similarity of the discrete and the singular case): Let $\gamma=\sum_{j\in
J}\alpha_{j}\delta_{a_{j}}$ for some finite our countably infinite index set
$J\subseteq\mathbb{N}$ where $\alpha_{j}>0$ for every $j\in J$, and
$\sum_{j\in J}\alpha_{j}=1$ (without loss of generality we assume $a_{i}\neq
a_{j}$ for $i\neq j$). Then for every $j\in J$ there exists a unique
$t_{j}\in[0,1)$ with $\frac{1}{\varphi(t_{j})}=a_{j}$ and according to Theorem
5.5 we have

$\mu_{C}\left(\bigcup_{j\in J}L_{t_{j}}\right)=\sum_{j\in
J}\mu_{C}(L_{t_{j}})=\sum_{j\in
J}\gamma\left(\left\\{\frac{1}{\varphi(t_{j})}\right\\}\right)=1.$

The set $L:=\bigcup_{j\in J}L_{t_{j}}$ is as at most countable union of Borel
sets itself an element of $\mathcal{B}(\mathbb{I}^{d})$. Applying
disintegration we have

$\displaystyle
1=\mu_{C}(L)=\int_{\mathbb{I}^{d-1}}K_{C}(\mathbf{x},L_{\mathbf{x}})\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{x}),$

from which $K_{C}(\mathbf{x},L_{\mathbf{x}})=1$ for $\mu_{C^{1:d-1}}$-almost
every $\mathbf{x}$ follows. Considering that the $\mathbf{x}$-cut
$L_{\mathbf{x}}$ of $L$ is at most countably infinite we get
$K_{C}^{dis}(\mathbf{x},L_{\mathbf{x}})=1$ for $\mu_{C^{1:d-1}}$-almost every
$\mathbf{x}$ from which the desired result follows.

Theorem 5.10 has the following consequence, whereby we will let
$\mathcal{C}_{ar,abs}^{d}$ denote the family of all absolutely continuous
$d$-dimensional Archimedean copulas, $\mathcal{C}_{ar,dis}^{d}$ the family of
all $C\in\mathcal{C}_{ar}^{d}$ with $\mu_{C}^{dis}(\mathbb{I}^{d})=1$, and
$\mathcal{C}_{ar,sing}^{d}$ the family of all $C\in\mathcal{C}_{ar}^{d}$ with
$\mu_{C}^{sing}(\mathbb{I}^{d})=1$.

######  Corollary 5.12.

$\mathcal{C}_{ar,dis}^{d},\,\mathcal{C}_{ar,abs}^{d}$ and
$\mathcal{C}_{ar,sing}^{d}$ are dense in $(\mathcal{C}_{ar}^{d},d_{\infty})$.

###### Proof.

Let $C$ be an arbitrary Archimedean copula and $\gamma$ denote its
corresponding Williamson measure on $\mathcal{B}([0,\infty))$. The stated
results now follow from the fact (see Theorem B.2) that $\gamma$ is the weak
limit of a sequence of discrete, of a sequence of absolutely continuous and of
a sequence of singular Williamson measures in combination with Theorem 5.8 and
Theorem 5.10. ∎

##  6\. Singular Archimedean copulas with full support

The results established in the previous section allow to prove the existence
of multivariate Archimedean copulas which, considering their handy analytic
form, exhibit a surprisingly irregular behavior. In fact, we will construct
singular $d$-dimensional Archimedean copulas with full support
$\mathbb{I}^{d}$ and thereby extend the examples given in [4] to the
multivariate setting. As in the previous section the representation in terms
of Williamson measures will play a crucial role. We first focus on the
construction of some $C\in\mathcal{C}_{ar}^{d}$ fulfilling that $C$ has full
support although $\mu_{C^{1:d-1}}$-almost every conditional distribution
$K_{C}(\mathbf{x},\cdot)$ is a singular measure without point masses and then
discuss the discrete analogue.

######  Theorem 6.1.

There exists a copula $C\in\mathcal{C}_{ar}^{d}$ with the following
properties:

  * (1)

$C$ is singular continuous and has full support.

  * (2)

For $\mu_{C^{1:d-1}}$-almost every $\mathbf{x}\in\mathbb{I}^{d-1}$ the
conditional distribution function $y\mapsto K_{C}(\mathbf{x},[0,y])$ is
continuous, strictly increasing and singular.

  * (3)

All level sets $L_{t}$ of $C$ fulfill $\mu_{C}(L_{t})=0$.

  * (4)

The Kendall distribution function $F_{K}^{d}$ of $C$ is continuous, strictly
increasing and singular.

###### Proof.

Suppose that $h$ is a strictly increasing singular homemorphism of
$\mathbb{I}$, i.e., a strictly increasing bijective transformation mapping
$\mathbb{I}$ to itself fulfilling $h^{\prime}(x)=0$ for $\lambda$-almost every
$x\in\mathbb{I}$ (see, e.g., [6, 10] for several well-known examples).
Defining $F:[0,\infty)\rightarrow[0,1]$ by

$F(x)=\frac{1}{2}\,h\left(\frac{x}{2}\right)\mathbf{1}_{[0,2)}(x)\,+\,\sum_{i=1}^{\infty}\left(1-\frac{1}{2^{i}}+\frac{1}{2^{i+1}}\,h\left(\frac{x-2^{i}}{2^{i}}\right)\right)\mathbf{1}_{[2^{i},2^{i+1})}(x)$

obviously yields a strictly increasing continuous function $F$ which, by
construction, fulfills $F^{\prime}=0$ $\lambda$-almost everywhere. Letting
$\beta$ denote the corresponding probability measure on
$\mathcal{B}([0,\infty))$ it follows that $\beta$ is singular without point
masses. Furthermore, the support of $\beta$ contains $0$ but in general does
not need to be an element of $\mathcal{P}_{\mathcal{W}_{d}}$, we only know
that

$\int_{\mathbb{I}}(1-t)^{d-1}\mathrm{d}\beta(t)\in(0,1).$

Proceeding, however, like in the proof of Lemma B.1 we can find some constant
$a\in(0,\infty)$ such that the push-forward $\gamma=\beta^{T_{a}}$ with
$T_{a}(x)=ax$ fulfills $\gamma\in\mathcal{P}_{\mathcal{W}_{d}}$. Considering
that $\gamma$ is obviously singular (without point masses) too and that the
support of $\gamma$ coincides with $[0,\infty)$ using Lemma 5.4 as well as
Theorem 5.10 it follows that the corresponding $d$-dimensional Archimedean
copula $C=C_{\gamma}$ is strict and fulfills
$\mu_{C_{\gamma}}^{sing}(\mathbb{I}^{d})=1$. Furthermore, according to the
proof of Theorem 5.10 for $\mu_{C^{1:d-1}}$-almost every
$\mathbf{x}\in\mathbb{I}^{d-1}$ the conditional distribution function
$y\mapsto K_{C}(\mathbf{x},[0,y])$ is continuous and singular. Hence,
considering that $\gamma$ has full support using eq. (5.11) yields that
$y\mapsto K_{C}(\mathbf{x},[0,y])$ is also strictly increasing on
$\mathbb{I}$.
Having that, showing that $C$ has full support is straightforward: In fact,
for every $(\mathbf{x},y)\in(0,1)^{d-1}\times(0,1)$ and every open rectangle
$U=U_{1}\times\cdots\times U_{d}$ with open non-empty intervals
$U_{1},\ldots,U_{d}\subseteq(0,1)$ fulfilling $(\mathbf{x},y)\in U$ and
$U\subseteq(0,1)^{d}$ we can proceed as follows: Considering that $C^{1:d-1}$
is absolutely continuous strictness of $\psi$ implies that the density
$c^{1:d-1}$ of $C^{1:d-1}$ fulfills $c^{1:d-1}>0$ $\lambda_{d-1}$-almost
everywhere in $(0,1)^{d-1}$ using disintegration and the fact that
$K_{C}(\mathbf{x},\cdot)$ has full support and hence fulfills
$K_{C}(\mathbf{x},U_{d})>0$ for $\mu_{C^{1:d-1}}$-almost every
$\mathbf{x}\in\mathbb{I}^{d-1}$ it follows that

$\displaystyle\mu_{C}(U)$
$\displaystyle=\int_{\times_{j=1}^{d-1}U_{j}}K_{C}(\mathbf{x},U_{d})\,\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{x})>0.$

This shows that $(\mathbf{x},y)$ is contained in the support of $\mu_{C}$,
since supports are closed the the support of $\mu_{C}$ is $\mathbb{I}^{d}$ and
the first two assertions are proved. Since $\gamma$ has no point masses the
third assertion is an immediate consequence of Theorem 5.5 and it remains to
prove the last assertion. Again according to Theorem 5.5

$F_{K}^{d}(t)=\gamma([0,\tfrac{1}{\varphi(t)}])=\gamma((0,\tfrac{1}{\varphi(t)}])$

holds for every $t\in(0,1]$, implying that $F_{K}^{d}$ is continuous and
strictly increasing. Finally, using a chain rule argument similar to the one
at the end of the proof of Theorem 5.10 shows that $(F_{K}^{d})^{\prime}(x)=0$
holds for $\lambda$-almost every $x\in\mathbb{I}$ and the proof is complete. ∎

Starting with the probability measure
$\beta:=\sum_{i=1}^{\infty}2^{-i}\delta_{q_{i}}$ with
$\\{q_{1},q_{2},\ldots\\}$ denoting an enumeration of the rationals in
$(0,\infty)$ and proceeding analogously to the proof of the previous theorem
yields the following discrete version of it:

######  Theorem 6.2.

There exists a copula $C\in\mathcal{C}_{ar}^{d}$ with the following
properties:

  * (1)

$C$ is singular continuous and has full support.

  * (2)

For $\mu_{C^{1:d-1}}$-almost every $\mathbf{x}\in\mathbb{I}^{d-1}$ the
conditional distribution function $y\mapsto K_{C}(\mathbf{x},[0,y])$ is a
strictly increasing step function.

  * (3)

There exists a dense countable subset $\mathcal{Q}$ of $(0,1)$ such that
$\mu_{C}(L_{t})>0$ if, and only if, $t\in\mathcal{Q}$.

  * (4)

The Kendall distribution function $F_{K}^{d}$ of $C$ is a strictly increasing
step function.

##  Appendix A Level set mass and Kendall distribution function: Calculations

Recall from Section 3 that the $t$-level hypersurfaces $f^{t}$ are defined on
the upper $t$-cuts $[C^{1:d-1}]_{t}$ of the $(d-1)$-marginal. Using the
notation $\mathbf{x}_{m}=(x_{1},x_{2},\ldots,x_{m})$, $m\in\mathbb{N}$, for
$\mathbf{x}\in[C^{1:d-1}]_{t}$ we have $x_{1}\geq
t,x_{2}\geq\psi(\varphi(t)-\varphi(x_{1}))=:f^{t}(x_{1}),x_{3}\geq\psi\left(\varphi(t)-\varphi(x_{1})-\varphi(x_{2})\right)=:f^{t}(\mathbf{x}_{2})$
and iteratively,

$x_{d-1}\geq\psi\big{(}\varphi(t)-\sum_{i=1}^{d-2}\varphi(x_{i})\big{)}=:f^{t}(\mathbf{x}_{d-2}).$

######  Proposition A.1.

Suppose that $C\in\mathcal{C}_{\text{ar}}^{d}$ has generator $\psi$ and let
$\mu_{C}$ denote the corresponding $d$-stochastic measure. Then for every
$t>0$ we have

(A.1)
$\displaystyle\mu_{C}(L_{t})=\frac{(-\varphi(t))^{d-1}}{(d-1)!}\cdot\big{(}D^{-}\psi^{(d-2)}(\varphi(t))-D^{-}\psi^{(d-2)}(\varphi(t-))\big{)}.$

If $C$ is strict then $\mu_{C}(L_{0})=0$ and for non-strict $C$,

(A.2) $\displaystyle\mu_{C}(L_{0})$
$\displaystyle=\frac{(-\varphi(0))^{d-1}}{(d-1)!}\cdot
D^{-}\psi^{(d-2)}(\varphi(0)).$

###### Proof.

We start with $t>0$. Using disintegration, the definition of $f^{t}$ and the
fact that $C^{1:d-1}$ is absolutely continuous with density $c^{1:d-1}$ we get

$\displaystyle\mu_{C}(L_{t})$
$\displaystyle=\int_{\mathbb{I}^{d-1}}K_{C}(\mathbf{s},(L_{t})_{\mathbf{s}})\
\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{s})$
$\displaystyle=\int_{[t,1]\times[f^{t}(s_{1}),1]\times\ldots\times[f^{t}(\mathbf{s}_{d-2}),1]}K_{C}(\mathbf{s},\\{f^{t}(\mathbf{s})\\})\
\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{s})$
$\displaystyle=\int_{[t,1]}\cdots\int_{[f^{t}(\mathbf{s}_{d-2}),1]}\prod_{i=1}^{d-1}\varphi^{\prime}(s_{i})\cdot\left[D^{-}\psi^{(d-2)}(\varphi(t))-D^{-}\psi^{(d-2)}(\varphi(t-))\right]\
\mathrm{d}\lambda(\mathbf{s})$
$\displaystyle=\left[D^{-}\psi^{(d-2)}(\varphi(t))-D^{-}\psi^{(d-2)}(\varphi(t-))\right]\cdot\int_{[t,1]}\cdots\int_{[f^{t}(\mathbf{s}_{d-2}),1]}\prod_{i=1}^{d-1}\varphi^{\prime}(s_{i})\
\mathrm{d}\lambda(\mathbf{s}).$

Letting $(II)$ denote the iterated integrals in the previous line we have

$\displaystyle(II)=\int\limits_{[t,1]}\int\limits_{[f^{t}(s_{1}),1]}\cdots\prod\limits_{i=1}^{d-2}\varphi^{\prime}(s_{i})\int\limits_{[f^{t}(\mathbf{s}_{d-2}),1]}\varphi^{\prime}(s_{d-1})(-1)^{0}\left[\varphi(t)-\sum\limits_{i=1}^{d-1-0}\varphi(s_{i})\right]^{0}\
\mathrm{d}\lambda(\mathbf{s})$

and the chain rule directly yields

$\displaystyle(II)$
$\displaystyle=\int_{[t,1]}\cdots\int_{[f^{t}(\mathbf{s}_{d-4}),1]}\prod\limits_{i=1}^{d-3}\varphi^{\prime}(s_{i})$
$\displaystyle\qquad\int_{[f^{t}(\mathbf{s}_{d-3}),1]}\varphi^{\prime}(s_{d-2})\cdot\frac{(-1)^{1}}{1}\cdot\left[\varphi(t)-\sum_{i=1}^{d-1-1}\varphi(s_{i})\right]^{1}\
\mathrm{d}\lambda(s_{d-2})\mathrm{d}\lambda(\mathbf{s}_{d-3}).$

Proceeding analogously for $s_{d-2}$ gives

$\displaystyle(II)$
$\displaystyle=\int_{[t,1]}\cdots\int_{[f^{t}(\mathbf{s}_{d-4}),1]}\prod_{i=1}^{d-3}\varphi^{\prime}(s_{i})\cdot\frac{(-1)^{2}}{1\cdot
2}\cdot\left[\varphi(t)-\sum_{i=1}^{d-1-2}\varphi(s_{i})\right]^{2}\
\mathrm{d}\lambda(\mathbf{s}_{d-3})$

and after finitely many steps we obtain

$\displaystyle(II)$
$\displaystyle=\int_{[t,1]}\varphi^{\prime}(s_{1})\cdot\frac{(-1)^{d-2}}{1\cdot
2\cdots(d-2)}\left[\varphi(t)-\varphi(s_{1})\right]^{d-2}\
\mathrm{d}\lambda(s_{1})=\frac{(-1)^{d-1}}{(d-1)!}\cdot\varphi(t)^{d-1}$

as desired. For $t=0$ and strict $C$ we obviously have $\mu_{C}(L_{0})=0$. For
non-strict $C$ we have
$K_{C}(\mathbf{s},\\{f^{0}(\mathbf{s})\\})=K_{C}(\mathbf{s},[0,f^{0}(\mathbf{s})])$
and calculations as those above yield the result. ∎

######  Proposition A.2.

Suppose that $C\in\mathcal{C}_{\text{ar}}^{d}$ has generator $\psi$. Then for
$t>0$

(A.3) $\displaystyle
F_{K}^{d}(t)=D^{-}\psi^{(d-2)}(\varphi(t))\frac{(-1)^{d-1}}{(d-1)!}\varphi(t)^{d-1}+\sum_{k=0}^{d-2}\psi^{(k)}(\varphi(t))\frac{(-1)^{k}}{k!}\varphi(t)^{k}.$

holds. For $t=0$ and strict $C$ we have $F_{K}^{d}(0)=0$ and for non-strict
$C$,

(A.4) $\displaystyle
F_{K}^{d}(0)=D^{-}\psi^{(d-2)}(\varphi(0))\cdot\frac{(-1)^{d-1}}{(d-1)!}\cdot\varphi(0)^{d-1}.$

###### Proof.

Applying disintegration and decomposing
$\mathbb{I}^{d-1}=[C^{1:d-1}]_{t}\cup[C^{1:d-1}]_{t}^{c}$ yields

$\displaystyle F_{K}^{d}(t)$
$\displaystyle=\mu_{C}([C]_{t}^{c})=\int_{\mathbb{I}^{d-1}}K_{C}\left(\mathbf{x},([C]_{t}^{c})_{\mathbf{x}}\right)\
\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{x})$
$\displaystyle=\int_{[C^{1:d-1}]_{t}}K_{C}\left(\mathbf{x},([C]_{t}^{c})_{\mathbf{x}}\right)\
\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{x})+\int_{[C^{1:d-1}]_{t}^{c}}K_{C}\left(\mathbf{x},([C]_{t}^{c})_{\mathbf{x}}\right)\
\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{x})$

Denoting by $(III)$ and $(IV)$ the first and the second of the above summands,
respectively, analogously as in Proposition A.1 we obtain

$\displaystyle(III)=D^{-}\psi^{(d-2)}(\varphi(t))\cdot\frac{(-1)^{d-1}}{(d-1)!}\varphi(t)^{d-1}.$

Regarding $(IV)$, we have $\mathbf{x}\in[C^{1:d-1}]_{t}^{c}$ if, and only if,
$([C]_{t}^{c})_{\mathbf{x}}=\mathbb{I}$ and hence

$\displaystyle(IV)$ $\displaystyle=\int_{[C^{1:d-1}]_{t}^{c}}1\
\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{x})=\mu_{C^{1:d-1}}([C^{1:d-1}]_{t}^{c})=F_{K}^{d-1}(t).$

Proceeding iteratively finally yields

$\displaystyle F_{K}^{d}(t)$
$\displaystyle=D^{-}\psi^{(d-2)}(\varphi(t))\cdot\frac{(-1)^{d-1}}{(d-1)!}\cdot\varphi(t)^{d-1}+\sum_{k=1}^{d-2}\psi^{(k)}(\varphi(t))\frac{(-1)^{k}}{k!}\varphi(t)^{k}+t.$

For $t=0$ we have $F_{K}^{d}(t)=\int_{\\{\mathbf{s}\in\mathbb{I}^{d-1}:\
\sum_{i=1}^{d-1}\varphi(s_{i})\leq\varphi(0)\\}}K_{C}(\mathbf{x},([C]_{0}^{c})_{\mathbf{x}})\
\mathrm{d}\mu_{C^{1:d-1}}(\mathbf{x})$ and the result follows in the same
manner. ∎

######  Lemma A.3.

Suppose that $f,f_{1},f_{2},\ldots$ are convex functions such that
$(f_{n})_{n\in\mathbb{N}}$ converges to $f$ pointwise on $(0,\infty)$. Then
$\lim_{n\rightarrow\infty}D^{-}f_{n}(x)=D^{-}f(x)$ holds for every
$x\in\text{Cont}(D^{-}f)$. Moreover, the sequence
$(D^{-}f_{n})_{n\in\mathbb{N}}$ converges continuously to $D^{-}f$ on
$\text{Cont}(D^{-}f)$, i.e., for every sequence $(z_{n})_{n\in\mathbb{N}}$
with limit $z\in\text{Cont}(D^{-}f)$ we have

$\lim_{n\rightarrow\infty}D^{-}f_{n}(z_{n})=D^{-}f(z).$

###### Proof.

Convexity implies that for every $h>0$ we have

$\displaystyle\limsup_{n\rightarrow\infty}D^{+}f_{n}(x)\leq\limsup_{n\rightarrow\infty}\frac{f_{n}(x+h)-f_{n}(x)}{h}=\frac{f(x+h)-f(x)}{h},$

which, considering $h\downarrow 0$, yields
$\limsup_{n\rightarrow\infty}D^{+}f_{n}(x)\leq D^{+}f(x)$. The inequality
$\liminf_{n\rightarrow\infty}D^{-}f_{n}(x)\geq D^{-}f(x)$ follows in the same
manner, so altogether we get

$\displaystyle
D^{-}f(x)\leq\liminf_{n\rightarrow\infty}D^{-}f_{n}(x)\leq\limsup_{n\rightarrow\infty}D^{+}f_{n}(x)\leq
D^{+}f(x)$

from which the fist assertion follows since for $x\in\text{Cont}(D^{-}f)$ we
have $D^{+}f(x)=D^{-}f(x)$.

For the second part let $(z_{n})_{n\in\mathbb{N}}$ be a sequence in
$(0,\infty)$ converging to some point
$z\in\text{Cont}(D^{-}f)\subseteq(0,\infty)$. We can find two strictly
decreasing sequences $(a_{k})_{k\in\mathbb{N}},(b_{k})_{k\in\mathbb{N}}$ in
$(0,\infty)$ converging to $0$ with $z-a_{k},z+b_{k}\in\text{Cont}(D^{-}f)$
for every $k\in\mathbb{N}$. Fix $k\in\mathbb{N}$. Then there exists some index
$n_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$ we have that
$z-a_{k}<z_{n}<z+b_{k}$ holds. Monotonicity of $D^{-}f_{n}$ implies

$\displaystyle D^{-}f_{n}(z-a_{k})\leq D^{-}f_{n}(z_{n})\leq
D^{-}f_{n}(z+b_{k})$

for every such $n\geq n_{0}$. Having this we get

$\displaystyle\lim_{n\rightarrow\infty}D^{-}f_{n}(z-a_{k})$
$\displaystyle\leq\liminf_{n\rightarrow\infty}D^{-}f_{n}(z_{n})$
$\displaystyle\leq\limsup_{n\rightarrow\infty}D^{-}f_{n}(z_{n})\leq\lim_{n\rightarrow\infty}D^{-}f_{n}(z+b_{k}),$

from which the result follows since $z\in\text{Cont}(D^{-}f)$ and therefore

$\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}D^{-}f_{n}(z-a_{k})=D^{-}f(z)=\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}D^{-}f_{n}(z+b_{k})$

holds. ∎

##  Appendix B Approximations by discrete, absolutely continuous and singular
Williamson measures

We now tackle Theorem B.2 already used in the proof of Corollary 5.12 and
start with the following simple lemma simplifying the approximation procedure.
Thereby, for every $a\in(0,\infty)$ we will let
$T_{a}:(0,\infty)\rightarrow(0,\infty)$ denote the linear transformation
$T_{a}(x)=ax$.

######  Lemma B.1.

Suppose that $\gamma\in\mathcal{P}_{\mathcal{W}_{d}}$, that
$(\beta_{n})_{n\in\mathbb{N}}$ is a sequence of probability measures on
$\mathcal{B}([0,\infty)$ satisfying $\beta_{n}(\\{0\\})=0$ for every
$n\in\mathbb{N}$ but not necessarily fulfilling eq. (5.2), and that
$(\beta_{n})_{n\in\mathbb{N}}$ converges weakly to $\gamma$. Then there exists
a sequence $(a_{n})_{n\in\mathbb{N}}$ in $(0,\infty)$ converging to
$\frac{1}{2}$ such that the following properties hold:

  * •

Each probability measure $\gamma_{n}:=\beta_{n}^{T_{a_{n}}},n\in\mathbb{N}$,
fulfills $\gamma_{n}\in\mathcal{P}_{\mathcal{W}_{d}}$.

  * •

$(\gamma_{n})_{n\in\mathbb{N}}$ converges weakly on $[0,\infty)$ to $\gamma$.

###### Proof.

Let $\psi_{\gamma}$ denote the normalized Archimedean generator corresponding
to $\gamma$ and $\psi_{\beta_{n}}$ the (not necessarily normalized) generator
induced via $\psi_{\beta_{n}}=\mathcal{W}_{d}(\beta_{n})$. Then proceeding as
in the first part of the proof of Theorem 5.8 it follows that
$(\psi_{\beta_{n}})_{n\in\mathbb{N}}$ converges uniformly to $\psi_{\gamma}$.
Letting $a_{n}$ denote the unique element in $(0,\infty)$ fulfilling
$\psi_{\beta_{n}}(a_{n})=\frac{1}{2}$ using monotonicity of generators and the
fact that $\psi_{\gamma}$ is normalized it is straightforward to verify that
$(a_{n})_{n\in\mathbb{N}}$ converges to $1$.
The probability measure $\gamma_{n}:=\beta_{n}^{T_{a_{n}}}$ obviously fulfills
$\gamma_{n}(\\{0\\})=0$, moreover using change of coordinates yields

$\displaystyle\frac{1}{2}$
$\displaystyle=\psi_{\beta_{n}}(a_{n})=\int_{(0,\infty)}(1-ta_{n})_{+}^{d-1}\,\mathrm{d}\beta_{n}(t)=\int_{(0,\infty)}(1-T_{a_{n}}(t))_{+}^{d-1}\,\mathrm{d}\beta_{n}(t)$
$\displaystyle=\int_{(0,\infty)}(1-s)_{+}^{d-1}\,\mathrm{d}\gamma_{n}(t)=\int_{(0,1)}(1-s)^{d-1}\,\mathrm{d}\gamma_{n}(t),$

so $\gamma_{n}\in\mathcal{P}_{\mathcal{W}_{d}}$ and it remains to show weak
convergence. Applying Lemma 5.2 and Lemma 4.4 (continuous convergence) yields

$\displaystyle\lim_{n\rightarrow\infty}\gamma_{n}([0,z])$
$\displaystyle=\lim_{n\rightarrow\infty}\beta_{n}\left(\left[0,\frac{z}{a_{n}}\right]\right)$
$\displaystyle=\lim_{n\rightarrow\infty}\sum_{k=0}^{d-2}\frac{(-1)^{k}\psi_{\beta_{n}}^{(k)}(\tfrac{a_{n}}{z})}{k!}\left(\frac{a_{n}}{z}\right)^{k}+\frac{(-1)^{d-1}D^{-}\psi_{\beta_{n}}^{(d-2)}(\tfrac{a_{n}}{z})}{(d-1)!}\left(\frac{a_{n}}{z}\right)^{d-1}$
$\displaystyle=\gamma([0,z])$

for every point $z\in(0,\infty)$ with $\gamma(\\{z\\})=0$ which completes the
proof. ∎

######  Theorem B.2.

Suppose that $\gamma\in\mathcal{P}_{\mathcal{W}_{d}}$. Then there exists a
sequence $(\gamma_{n}^{1})_{n\in\mathbb{N}}$ of discrete measures in
$\mathcal{P}_{\mathcal{W}_{d}}$, a sequence
$(\gamma_{n}^{2})_{n\in\mathbb{N}}$ of singular measures without point masses
in $\mathcal{P}_{\mathcal{W}_{d}}$, and a sequence
$(\gamma_{n}^{3})_{n\in\mathbb{N}}$ of absolutely continuous measures
$\mathcal{P}_{\mathcal{W}_{d}}$ that all converge weakly to $\gamma$ on
$[0,\infty)$.

###### Proof.

Let $F$ denote the distribution function corresponding to $\gamma$, i.e.,
$F(z)=\gamma((0,z])=\gamma([0,z])$ for every $z\in[0,\infty)$ and let
$\mathcal{Q}:=\\{q_{0},q_{1},q_{2},\ldots\\}$ denote a countably infinite
subset of Cont$(F)$ which is dense in $[0,\infty)$. Without loss of generality
we assume that $q_{0}=0$. Furthermore let the function
$f:\mathbb{I}\rightarrow[0,1]$ be right-continuous, non-decreasing with
$f(0)=0,f(1)=1$ and $g:[0,\infty)\rightarrow\mathbb{I}$ be right-continuous,
non-decreasing with $g(0)=0,g(\infty)=1$. For every non-degenerated compact
interval $[a,b]\subseteq[0,\infty)$ and compact interval $[c,d]\subseteq[0,1]$
define the rescaled version $f_{[a,b]}^{[c,d]}:[a,b]\rightarrow[c,d]$ of $f$
to $[a,b],[c,d]$ by

$f_{[a,b]}^{[c,d]}(x)=c+(d-c)f\left(\frac{x-a}{b-a}\right),$

and for every interval $[a,\infty)\subseteq[0,\infty)$ and
$[c,d]\subseteq\mathbb{I}$ define
$g_{[a,\infty)}^{[c,d]}:[a,\infty)\rightarrow[c,d]$ of $g$ to
$[a,\infty),[c,d]$ by

$g_{[a,\infty)}^{[c,d]}(x)=c+(d-c)g(x-a).$

Using this notation define the distribution function
$F_{1}:[0,\infty)\rightarrow[0,1]$ by

$F_{1}(x)=f_{[0,q_{1}]}^{[0,F(q_{1})]}(x)\cdot\mathbf{1}_{[0,q_{1}]}(x)\,+\,g_{[q_{1},\infty)}^{[F(q_{1}),1]}(x)\cdot\mathbf{1}_{(q_{1},\infty)}(x)$

and notice that $F_{1}$ fulfills $F_{1}(q_{i})=F(q_{i})$ for $i\in\\{0,1\\}$.
In the second step define the distribution function
$F_{2}:[0,\infty)\rightarrow[0,1]$ by

$F_{2}(x)=f_{[0,q^{2}_{(1)}]}^{[0,F(q^{2}_{(1)})]}(x)\cdot\mathbf{1}_{[0,q^{2}_{(1)}]}(x)\,+\,f_{[q^{2}_{(1)},q^{2}_{(2)}]}^{[F(q^{2}_{(1)}),F(q^{2}_{(2)})]}(x)\cdot\mathbf{1}_{(q^{2}_{(1)},q^{2}_{(2)}]}(x)\,+\,g_{[q^{2}_{(2)},\infty)}^{[F(q^{2}_{(2)}),1]}(x)\cdot\mathbf{1}_{(q^{2}_{(2)},\infty)}(x),$

whereby $0<q^{2}_{(1)}<q^{2}_{(2)}$ denotes the order statistics of
$0,q_{1},q_{2}$ (the exponent denotes the ‘sample size’). Obviously $F_{2}$
fulfills $F_{2}(q_{i})=F(q_{i})$ for $i\in\\{0,1,2\\}$. Proceeding analogously
yields a sequence $(F_{n})_{n\in\mathbb{N}}$ of distribution functions on
$[0,\infty)$ fulfilling that for every $i\in\mathbb{N}$ we have
$F_{n}(q_{i})=F(q_{i})$ for every $n\geq i$. In other words, the sequence
$(F_{n})_{n\in\mathbb{N}}$ converges on a dense set to the distribution
function $F$, implying that $(F_{n})_{n\in\mathbb{N}}$ converges to $F$
weakly.
Notice that the just discussed construction works for arbitrary $f,g$
fulfilling the afore-mentioned requirements. If we chose both $f$ and $g$
absolutely continuous then obviously each $F_{n}$ is absolutely continuous, if
we chose both $f$ and $g$ as step functions then $F_{n}$ is a step function,
and if we choose $f,g$ to be continuous with $f^{\prime}=0$ and $g^{\prime}=0$
almost everywhere (one could use, for instance, the Cantor function or work
with any other strictly increasing singular continuous distribution function,
see [10]) then each $F_{n}$ is continuous and obviously fulfills
$F_{n}^{\prime}=0$ almost everywhere. The desired result now follows by
considering the probability measures $\beta_{n}$ corresponding to $F_{n}$ and
applying Lemma B.1. ∎

## Acknowledgements

The first author gratefully acknowledges the financial support from Porsche
Holding Austria and Land Salzburg within the WISS 2025 project ‘KFZ’
(P1900123). The second and the third author gratefully acknowledge the support
of the WISS 2025 project ‘IDA-lab Salzburg’ (20204-WISS/225/197-2019 and
20102-F1901166-KZP).

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