# Sharpening the probabilistic
Arithmetic-Geometric Mean Inequality

Tom Kaufmann  and  Christoph Thäle  Tom Kaufmann: Faculty of Mathematics, Ruhr
University Bochum, Germany<EMAIL_ADDRESS>Christoph Thäle: Faculty of
Mathematics, Ruhr University Bochum, Germany<EMAIL_ADDRESS>
###### Abstract.

We consider the $p$-generalized arithmetic-geometric mean inequality for
vectors chosen randomly from the $\ell_{p}^{n}$-ball in $\mathbb{R}^{n}$. In
this setting the inequality can be improved or reversed up to a respective
scalar constant with high probability, and central limit theorems and large
deviation results with respect to this constant have been shown. We sharpen
these large deviation results in the spirit of Bahadur and Ranga Rao, thereby
providing concrete and asymptotically exact estimates on a non-logarithmic
scale for the probability of the inequality being improvable or reversible up
to a constant, respectively.

###### Key words and phrases:

Arithmetic-geometric mean inequality, high-dimensional convex geometry,
$\ell_{p}^{n}$-ball, reverse inequality, sharp large deviations

###### 2010 Mathematics Subject Classification:

46B09, 52A23, 60F10

##  1\. Introduction and Main Results

For $n\in\mathbb{N}$ and $x_{1},\ldots,x_{n}\in\mathbb{R}^{n}$ the arithmetic-
geometric mean (AGM) inequality states that

${\left(\prod_{i=1}^{n}|x_{i}|\right)}^{1/n}\leq\quad{\frac{1}{n}\sum_{i=1}^{n}|x_{i}|}.$

Additionally, for $p>0$ the $p$-generalized arithmetic-geometric mean
($p$-AGM) inequality expands the above for the $p$-generalized mean, i.e. for
$(x_{1},\ldots,x_{n})\in\mathbb{R}^{n},n\in\mathbb{N}$, we have

${\left(\prod_{i=1}^{n}|x_{i}|\right)}^{1/n}\leq\quad{\left(\frac{1}{n}\sum_{i=1}^{n}|x_{i}|^{p}\right)}^{1/p}.$
(1)

It was shown by Gluskin and Milman [9] that for a random vector
$X^{(n)}\in\mathbb{R}^{n}$ uniformly distributed on the standard
$(n-1)$-dimensional unit sphere ${\mathbb{S}}^{n-1}$ in $\mathbb{R}^{n}$, one
can reverse the $p$-AGM inequality in (1) for $p=2$ up to a scalar constant
with high probability, which was then extended to $p=1$ by Aldaz [1, 2].
Kabluchko, Prochno and Vysotsky [13] provided a central limit theorem (CLT)
and a large deviation principle (LDP) for the ratio of the two sides of the
$p$-AGM inequality for any $p\in[1,\infty)$ and $X^{(n)}$ uniformly
distributed within the $\ell_{p}^{n}$-ball ${\mathbb{B}}_{p}^{n}$ or
distributed according to the surface measure or the cone probability measure
on the $\ell_{p}^{n}$-sphere ${\mathbb{S}}_{p}^{n-1}$, where

${\mathbb{B}}_{p}^{n}:=\\{x\in\mathbb{R}^{n}:\|x\|_{p}\leq 1\\}\text{ \qquad
and \qquad}{\mathbb{S}}_{p}^{n-1}:=\\{x\in\mathbb{R}^{n}:\|x\|_{p}=1\\},$

with

$\|x\|_{p}:=\Big{(}\sum\limits_{i=1}^{n}|x_{i}|^{p}\Big{)}^{1/p}.$

The cone probability measure on ${\mathbb{S}}_{p}^{n-1}$ is defined as

$\mathbf{C}_{n,p}(\,\cdot\,):={\mathrm{vol}_{n}(\\{rx:r\in[0,1],x\in\,\cdot\,\\})\over\mathrm{vol}_{n}({\mathbb{B}}_{p}^{n})},$

where $\mathrm{vol}_{n}$ denotes the $n$-dimensional Lebesgue measure on
$\mathbb{R}^{n}$. Finally, Thäle [21] then expanded the results of [13] to a
CLT and a moderate deviation principle (MDP) for the ratio of the two sides of
the $p$-AGM inequality with the corresponding random vector
$X^{(n)}\in{\mathbb{B}}_{p}^{n}$ having a distribution from a wider class of
$p$-radial distributions, established by Barthe, Guédon, Mendelson and Naor in
[5], which includes the uniform distribution and the cone probability measure
as special cases. This class of distributions is constructed by mixing the
uniform distribution and the cone probability measure via a $p$-radial
density, which is given by an additional distribution on $[0,\infty)$.
However, the arguments of Thäle show that the properties of interest of a
random vector $X^{(n)}\in{\mathbb{B}}^{n}_{p}$ are independent of the
$p$-radial component of its distribution, as long as the directional
distribution is given by $\mathbf{C}_{n,p}$ and its $p$-radial distribution
has no atom at zero (see Section 2). This means, that there is a distribution
$\mathbf{R}$ on $[0,1]$ with $\mathbf{R}(\\{0\\})=0$, such that for a random
variable $R$ with distribution $\mathbf{R}$ and a random variable $Z^{(n)}$
with distribution $\mathbf{C}_{n,p}$ that is independent of $R$, we have that

$X^{(n)}\overset{d}{=}R\cdot Z^{(n)},$ (2)

where $\overset{d}{=}$ denotes equality in distribution. (This can also be
expanded to sequences of $p$-radial distributions
$\big{(}\mathbf{R}^{(n)}\big{)}_{n_{\mathbb{N}}}$, if the limiting
distribution also has no atom at zero). Thus, it follows from [21] that the
same CLT and MDP hold universally for the ratio of the two sides of the
$p$-AGM inequality for any random vector $X^{(n)}\in{\mathbb{B}}^{n}_{p}$ with
directional distribution $\mathbf{C}_{n,p}$.

The purpose of this paper is to develop further the large deviation results of
[13] into sharp large deviations (SLD) results in the spirit of Bahadur and
Ranga Rao [4]. The theory of sharp large deviations has only very recently
been introduced into the field of high-dimensional convex geometry, starting
with Liao and Ramanan [16] and followed up by Kaufmann [14], both in the
context of $\ell_{p}^{n}$-balls and -spheres. It has the distinct advantage
over classical large deviations theory that it gives tail asymptotics not on a
logarithmic scale and can provide concrete and asymptotically exact tail
estimates for specific $n\in\mathbb{N}$. Moreover, just as LDP results are
more sensitive to the underlying distributions than e.g. CLTs and MDPs, SLD
results are so to an even greater extent than LDPs. For the sake of brevity,
we will not recapitulate the basics of (sharp) large deviations theory here,
and defer the reader to the works [14, 16] for an overview of the background
in the current setting of $\ell_{p}^{n}$-balls, and to the classic literature
[4, 7, 8, 17] for a more detailed account of the relevant theory.

For a random vector $X^{(n)}\in{\mathbb{B}}_{p}^{n}$ with directional
distribution $\mathbf{C}_{n,p}$ in the sense of (2) we now want to give sharp
asymptotics for the probability of the ratio of the two sides of the $p$-AGM
inequality in (1) being bigger than a constant $\theta\in[0,1]$. To state our
main result, we need to define the following functions: For
$\tau=(\tau_{1},\tau_{2})\in\mathbb{R}^{2}$, set

$\Lambda_{p}(\tau):=\log\left(\frac{1}{2p^{1/p}\Gamma\big{(}1+\frac{1}{p}\big{)}}\int_{\mathbb{R}}e^{\tau_{1}\log(|y|)+\frac{1}{p}(\tau_{2}-1)|y|^{p}}\,\textup{d}y\right),$

and for $x\in\mathbb{R}^{2}$ denote the Legendre-Fenchel transform of
$\Lambda_{p}$ as

$\displaystyle\Lambda_{p}^{*}(x):=\sup_{\tau\in\mathbb{R}^{2}}\big{[}\langle
x,\tau\rangle-\Lambda_{p}(\tau)\big{]},$

where $\langle\,\cdot\,,\,\cdot\,\rangle$ denotes the standard scalar product
in $\mathbb{R}^{2}$. We denote by $\mathcal{J}_{p}$ the effective domain of
$\Lambda_{p}^{*}$, i.e. the set of arguments for which $\Lambda_{p}^{*}$ is
finite. For an $x\in\mathcal{J}_{p}$, we denote by $\tau(x)\in\mathbb{R}^{2}$
the coefficients in the above term, where the supremum is attained, i.e. where
it holds that

$\Lambda_{p}^{*}(x)=\langle x,\tau(x)\rangle-\Lambda_{p}(\tau(x)).$

For a detailed argument for the existence of $\tau(x)$, we refer the reader to
[14, p. 246 f.], particularly the explanation regarding Equation (9) therein.
For a function $g:\mathbb{R}^{d}\to\mathbb{R}^{d}$, we denote by
$J_{x}g(x^{*})$ the Jacobian of $g$ with respect to the vector $x$ evaluated
at $x^{*}\in\mathbb{R}^{d}$, and for $f:\mathbb{R}^{d}\to\mathbb{R}$ by
$\nabla_{x}f(x^{*})$ and $\mathcal{H}_{x}f(x^{*})$ the gradient and Hessian of
$f$ with respect to the vector $x$ evaluated at $x^{*}\in\mathbb{R}^{d}$,
respectively. Moreover, let

$\psi(x)=\frac{\Gamma^{\prime}(x)}{\Gamma(x)},\qquad\text{with }x>0,$

be the digamma function. We then set

$m_{p}:=\frac{1}{p}\left(\psi\left(\frac{1}{p}\right)+\log(p)\right)<0.$

As we will see in Section 2, $e^{m_{p}}$ is the limit towards which the
expectations of the ratio of the $p$-AGM inequality converge in
$n\in\mathbb{N}$. Furthermore, we need to define the functions $\xi(\theta)$
and $\kappa(\theta)$ for $\theta\in[0,1]$, as used also in the sharp large
deviation results of Bahadur and Ranga Rao [4]. For $x\in\mathbb{R}^{2}$, we
set

$\displaystyle\mathfrak{H}_{x}:=\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))$ (3)

to be the Hessian of $\Lambda_{p}(\tau)$ in $\tau\in\mathbb{R}^{2}$, evaluated
at $\tau(x)$. For $\theta\in(0,1]$, we denote
$\theta^{*}:=(\log\theta,1)\in\mathbb{R}^{2}$ and for $\theta\in(0,1)$ we
define

$\displaystyle\xi(\theta)^{2}:=\langle\mathfrak{H}_{\theta^{*}}\,\tau(\theta^{*}),\tau(\theta^{*})\rangle\,\det\mathfrak{H}_{\theta^{*}},$

and

$\displaystyle\kappa(\theta)^{2}:=1-c_{\kappa}(\theta),$

with $c_{\kappa}(\theta)$ given by

$\frac{\left(\tau(\theta^{*})_{1}^{2}+\tau(\theta^{*})_{2}^{2}\right)^{3/2}\,p^{2}e^{p\theta}\theta^{-p}}{\big{|}\tau(\theta^{*})_{2}^{2}\left(\mathfrak{H}_{\theta^{*}}^{-1}\right)_{11}-2\tau(\theta^{*})_{1}\tau(\theta^{*})_{2}\left(\mathfrak{H}_{\theta^{*}}^{-1}\right)_{12}+\tau(\theta^{*})_{1}^{2}\left(\mathfrak{H}_{\theta^{*}}^{-1}\right)_{22}\big{|}\,\left(1+p^{2}e^{2p\theta}\theta^{-2p}\right)^{3/2}}.$

In the following results and throughout this paper, we denote by $o(1)$ a
sequence that tends to zero as $n$ tends to $\infty$. With the necessary
definitions and notation set up, we now proceed to formulate our main result.

######  Theorem 1.1.

Let $1\leq p<\infty$, $n\in\mathbb{N}$, and $X^{(n)}$ be a random vector in
${\mathbb{B}}^{n}_{p}$ with directional distribution $\mathbf{C}_{n,p}$ in the
sense of (2). It then holds

  1. i)

for $\theta\in(e^{m_{p}},1)$ and $n$ sufficiently large that

$\mathbb{P}\Big{[}\Big{(}\prod_{i=1}^{n}|X_{i}^{(n)}|\Big{)}^{1/n}>\theta\cdot\Big{(}\sum_{i=1}^{n}|X_{i}^{(n)}|\Big{)}^{1/p}\Big{]}=\displaystyle\frac{1}{\sqrt{2\pi
n}\,\kappa(\theta)\xi(\theta)}\,e^{-n\,\mathcal{I}_{p}(\theta)}\,(1+o(1)),$

  2. ii)

and for $\theta\in(0,e^{m_{p}})$ and $n$ sufficiently large that

$\mathbb{P}\Big{[}\Big{(}\prod_{i=1}^{n}|X_{i}^{(n)}|\Big{)}^{1/n}<\theta\cdot\Big{(}\sum_{i=1}^{n}|X_{i}^{(n)}|\Big{)}^{1/p}\Big{]}=\displaystyle\frac{1}{\sqrt{2\pi
n}\,\kappa(\theta)\xi(\theta)}\,e^{-n\,\mathcal{I}_{p}(\theta)}\,(1+o(1)),$

where

$\displaystyle\mathcal{I}_{p}(\theta)$ $\displaystyle:=$
$\displaystyle[pG_{p}(\theta)-1]\log(\theta)+G_{p}(\theta)[\log
G_{p}(\theta)-1]-\log\Gamma(G_{p}(\theta))$
$\displaystyle+\frac{1}{p}(1+\log(p))+\log\Gamma\left(\frac{1}{p}\right),$

with $G_{p}(\theta):=H^{-1}(p\log(\theta))$, where
$H:(0,\infty)\to(-\infty,0)$ is an increasing bijection given by

$H(x):=\psi(x)-\log(x).$ (4)

The two parts of the above theorem describe the decay of the probabiliy that
the $p$-AGM inequality is either reversible with a prefactor
$\theta\in(e^{m_{p}},1)$ [part i)] or can be sharpened with a prefactor
$\theta\in(0,e^{m_{p}})$ [part ii)]. Conversely, their respective opposites,
i.e. the probabilities that the inequality can be reversed with a prefactor
$\theta\in(0,e^{m_{p}})$ or sharpened with a prefactor
$\theta\in(e^{m_{p}},1)$ tend to 1 in $n\in\mathbb{N}$. This will be pointed
out in further detail in Section 2.

Note that the rate function $\mathcal{I}_{p}$ is not dependent on the
$p$-radial distribution of $X^{(n)}$, as is also the case in [13, 21], even
though SLD results usually tend to be more sensitive to the idiosyncrasies of
the underlying distributions.

These results are consistent with the large deviation principle of Kabluchko,
Prochno, and Vysotsky, as taking the logarithm of the probability in the above
theorem, dividing by $n$, and then considering the limit, yields what they
have shown in [13, Theorem 1.2], namely that

$\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}\Big{[}\Big{(}\prod_{i=1}^{n}|X_{i}^{(n)}|\Big{)}^{1/n}>\theta\cdot\Big{(}\sum_{i=1}^{n}|X_{i}^{(n)}|\Big{)}^{1/p}\Big{]}=-\mathcal{I}_{p}(\theta)$
(5)

for $\theta\in(e^{m_{p}},1)$ and

$\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}\Big{[}\Big{(}\prod_{i=1}^{n}|X_{i}^{(n)}|\Big{)}^{1/n}>\theta\cdot\Big{(}\sum_{i=1}^{n}|X_{i}^{(n)}|\Big{)}^{1/p}\Big{]}=-\mathcal{I}_{p}(\theta)$
(6)

for $\theta\in(0,e^{m_{p}})$. However, we do provide a refinement of their
results, since Theorem 1.1 gives estimates on a non-logarithmic scale and we
can thereby give concrete and asymptotically exact probability estimates for
the reversibility and improvability of the $p$-AGM inequality for a specific
(sufficiently large) $n\in\mathbb{N}$, whereas the prefactor in Theorem 1.1
vanishes on the logarithmic scale of a large deviation principle as in (5) and
(6). The proof of both Theorem 1.1 will follow closely along the lines of
Kaufmann [14] (and we defer to the proofs therein, where the arguments are
analogue) and is structured in three steps, each of which will have a
dedicated section. In Section 2 the ratio of the two sides of the $p$-AGM
inequality, denoted as $\mathcal{R}_{n}$, will be reformulated in terms of so-
called $p$-generalized Gaussian random vectors via well-established
representation results of Rachev and Rüschendorf [19] and Schechtmann and Zinn
[20]. Furthermore, the large deviation results of Kabluchko, Prochno, and
Vysotsky [13] for $\mathcal{R}_{n}$ will be given explicitly and expanded to
general distributions with directional component $\mathbf{C}_{n,p}$. In
Section 3 we will provide a local density approximation for this probabilistic
representation and then prove the main result in Section 4 by integrating over
the density estimate. For the latter, a geometric result for Laplace integral
expansions from Adriani and Baldi [3] is utilized.

##  2\. Probabilistic Representation

For a random vector $X^{(n)}\in{\mathbb{B}}^{n}_{p}$ with directional
distribution $\mathbf{C}_{n,p}$ in the sense of (2) the main variable of
interest is the ratio of the two sides of the $p$-AGM inequality given as

$\mathcal{R}_{n}:=\frac{{\left(\prod_{i=1}^{n}|X_{i}^{(n)}|\right)}^{1/n}}{{\left(\frac{1}{n}\sum_{i=1}^{n}|X_{i}^{(n)}|^{p}\right)}^{1/p}}.$
(7)

We want to formulate the target probabilities
$\mathbb{P}(\mathcal{R}_{n}>\theta)$ and $\mathbb{P}(\mathcal{R}_{n}<\theta)$
via a random vector $Y^{(n)}$ with generalized Gaussian distribution. In
general, we say a real-valued random variable $X$ has a generalized Gaussian
distribution if its distribution has Lebesgue density

$\displaystyle
f_{\textup{gen}}(x):=\displaystyle\frac{b}{2\,a\,\Gamma\left(\frac{1}{b}\right)}\,e^{-\big{(}{|x-m|}/{a}\big{)}^{b}},\qquad
x\in\mathbb{R},$

where $m\in\mathbb{R}$ and $a,b>0$, and denote this by
$X\sim{\mathbf{N}}_{\textup{gen}}(m,a,b)$. For our probabilistic
representation, we will specifically use the generalized Gaussian distribution
$\mathbf{N}_{p}:={\mathbf{N}}_{\textup{gen}}\left(0,p^{1/p},p\right)$,
$p\in[1,\infty),$ often referred to as a $p$-generalized Gaussian
distribution, with density

$\displaystyle
f_{p}(x):=\frac{1}{2\,p^{1/p}\,\Gamma\big{(}1+\frac{1}{p}\big{)}}\,e^{-{|x|^{p}}/{p}},\qquad
x\in\mathbb{R}.$

With this, we have the following useful representation resultfor the cone
probability measure $\mathbf{C}_{n,p}$ shown in [19] and [20].

######  Proposition 2.1.

Let $1\leq p<\infty$, $Y=(Y_{1},\ldots,Y_{n})$ be a random vector with
$Y_{i}\sim\mathbf{N}_{p}$ i.i.d., then the random vector ${Y}/{\|Y\|_{p}}$ has
distribution $\mathbf{C}_{n,p}$ and is independent of $\|Y\|_{p},$.

######  Remark 2.2.

Consider a random vector $X^{(n)}\in{\mathbb{B}}^{n}_{p}$ with directional
distribution $\mathbf{C}_{n,p}$ and $p$-radial distribution $\mathbf{R}$ on
$[0,1]$ in the sense of (2). If $\mathbf{R}$ is the Dirac measure at $1$, the
overall distribution of $X^{(n)}$ is again the cone measure
$\mathbf{C}_{n,p}$. Choosing $\mathbf{R}$ to be a beta distribution
$\mathbf{B}(\frac{n}{p},1)$ causes $X^{(n)}$ to be uniformly distributed in
${\mathbb{B}}^{n}_{p}$. Finally, for $m\in\mathbb{N}$, setting $\mathbf{R}$ to
be a beta distribution $\mathbf{B}(\frac{n}{p},\frac{m}{p})$, the distribution
of $X^{(n)}$ then corresponds to the orthogonal projection of
$\mathbf{C}_{n+m,p}$ on ${\mathbb{B}}^{n+m}_{p}$ onto its first $n$
coordinates, which for $p=m$ interestingly yields the uniform distribution on
${\mathbb{B}}^{n}_{p}$. All of these identities follow from [5, Theorem 1 & 2,
Corollary 3] by calculating the $p$-radial distributions of the random
variables considered therein (see also [18, Section 3]).

It directly follows from Proposition 2.1 for a random vector
$X^{(n)}\in{\mathbb{B}}^{n}_{p}$ with directional distribution
$\mathbf{C}_{n,p}$ and $p$-radial distribution $\mathbf{R}$ on $[0,1]$ in the
sense of (2) that

$\displaystyle\mathcal{R}_{n}\overset{d}{=}\frac{{\left(\prod\limits_{i=1}^{n}\left|R\,\frac{Y_{i}^{(n)}}{\|Y_{i}^{(n)}\|_{p}}\right|\right)}^{1/n}}{{\left(\frac{1}{n}\sum\limits_{i=1}^{n}\left|R\,\frac{Y_{i}^{(n)}}{\|Y_{i}^{(n)}\|_{p}}\right|^{p}\right)}^{1/p}}=\frac{{\left(\prod\limits_{i=1}^{n}\left|Y_{i}^{(n)}\right|\right)}^{1/n}}{{\left(\frac{1}{n}\sum\limits_{i=1}^{n}\left|Y_{i}^{(n)}\right|^{p}\right)}^{1/p}}.$
(8)

Thus, we see that $\mathcal{R}_{n}$ does not depend on the $p$-radial
distribution $\mathbf{R}$, which is why the rate function in the main result
is universal for all random vectors in ${\mathbb{B}}^{n}_{p}$ with directional
distribution $\mathbf{C}_{n,p}$. This calculation also shows that the CLT and
LDP established in [13] and the MDP shown by Thäle [21] also hold for any
random vector in ${\mathbb{B}}^{n}_{p}$ with directional distribution
$\mathbf{C}_{n,p}$ in the sense of (2). In the light of the above argument in
(8), let us present the LDP based on [13, Theorem 1.4] here in this more
general form.

######  Proposition 2.3.

Let $1\leq p<\infty$ and $X^{(n)}$ be a random vector in
${\mathbb{B}}_{p}^{n}$ with directional distribution $\mathbf{C}_{n,p}$ in the
sense of (2). Then the sequence $(\mathcal{R}_{n})_{n\in\mathbb{N}}$ with
$\mathcal{R}_{n}$ as defined in (7) based on $X^{(n)}$ satisfies an LDP on
$[0,1]$ with speed $n$ and rate function $\mathcal{I}_{p}$ as in Theorem 1.1.

It is furthermore shown in [13] that $\mathcal{I}_{p}(e^{m_{p}})=0$ and
$\mathcal{I}_{p}(0+)=\mathcal{I}_{p}(1-)=+\infty$, where $\mathcal{I}_{p}(0+)$
and $\mathcal{I}_{p}(1-)$ denote the limits of $\mathcal{I}_{p}$ to for
sequences that converge to $0$ and $1$ from above and below, respectively. As
suggested by the central limit result in [13, Theorem 1.1], the expectations
of $\mathcal{R}_{n}$ converge to $e^{m_{p}}$, i.e. the value from which
deviation probabilities are given in the above LDP and by the SLD results in
this paper.

For the $p$-AGM inequality this means two things: On the one hand, for
sufficiently large $n\in\mathbb{N}$, it can be reversed with prefactors in
$(0,e^{m_{p}})$ with high probability, and for prefactors in $(e^{m_{p}},1)$
the decay of this “reversion probability” is described by the rate function in
Theorem 1.1 i). On the other hand, for sufficiently large $n\in\mathbb{N}$, it
can be improved with prefactors in $(e^{m_{p}},1)$ with high probability, with
the decay of this “improvement probability” for prefactors in $(0,e^{m_{p}})$
being described by the rate function in Theorem 1.1 ii).

Proposition 2.3 is proven in [13] by showing an LDP for the sequence of
empirical averages of the coordinates of the random vector

$V^{(n)}:=\left(V_{1}^{(n)},\ldots,V_{n}^{(n)}\right),\qquad\text{with}\qquad
V_{i}^{(n)}:=\left(\log|Y_{i}^{(n)}|,|Y_{i}^{(n)}|^{p}\right),$ (9)

with $Y_{i}^{(n)}$ i.i.d. and $Y_{i}^{(n)}\sim\mathbf{N}_{p}$. This is done
via Cramér’s theorem (see e.g. [7, Theorem 2.2.30, Theorem 6.1.3, Corollary
6.1.6]), i.e. by showing that the cumulant generating function
$\Lambda_{V}(\tau):=\mathbb{E}e^{\langle\tau,V_{i}^{(n)}\rangle},\tau\in\mathbb{R}^{2}$
of the $V_{i}^{(n)}$ is finite in a neighbourhood of the origin, hence the
sequence of empirical averages of the coordinates

$S^{(n)}:=\frac{1}{n}\sum\limits_{i=1}^{n}V_{i}^{(n)}=\frac{1}{n}\sum\limits_{i=1}^{n}\left(\log|Y_{i}^{(n)}|,|Y_{i}^{(n)}|^{p}\right)$
(10)

satisfies an LDP with speed $n$ and rate function $\Lambda_{V}^{*}$, which is
the Legendre-Fenchel transform of $\Lambda_{V}$. Note that
$\Lambda_{V}=\Lambda_{p}$ and $\Lambda_{V}^{*}=\Lambda_{p}^{*}$. This LDP is
then mapped to the sequence $(\mathcal{R}_{n})_{n\in\mathbb{N}}$ via the
representation result from Proposition 2.1 and the contraction principle (see
e.g. [7, Theorem 4.2.1]), considering the map
$F(x_{1},x_{2}):=e^{x_{1}}{x_{2}}^{-1/p}$, yielding an LDP for
$(\mathcal{R}_{n})_{n\in\mathbb{N}}$ with speed $n$ and rate function

$\inf\limits_{(x_{1},x_{2}):\,F(x_{1},x_{2})=\theta}\,\Lambda_{p}^{*}(x_{1},x_{2}),\qquad\theta\in[0,1].$

This is then finalized by showing that the above infimum is attained uniquely
at $\theta^{*}:=(\log\theta,1)$ and that this infimum can be given explicitly
as

$\inf\limits_{(x_{1},x_{2}):\,F(x_{1},x_{2})=\theta}\,\Lambda_{p}^{*}(x_{1},x_{2})=\Lambda_{p}^{*}(\theta^{*})=\mathcal{I}_{p}(\theta).$
(11)

Further, it is shown that the effective domain of $\mathcal{I}_{p}$ is $(0,1)$
and that for $x\in\mathcal{J}_{p}$ it holds that

$\tau(x)=\Big{(}p\,H^{-1}(px_{1}-\log
x_{2})-1,\frac{1}{p}-x_{2}^{-1}\,H^{-1}(px_{1}-\log x_{2})\Big{)}$ (12)

with $H$ as in (4) (see [13, p. 11 f.]). We will use the same probabilistic
representations from (9) and (10), but proceed with them in a different
fashion. It holds that

$\mathbb{P}(\mathcal{R}_{n}>\theta)=\mathbb{P}(S^{(n)}\in
D_{\theta,>})\quad\text{and}\quad\mathbb{P}(\mathcal{R}_{n}<\theta)=\mathbb{P}(S^{(n)}\in
D_{\theta,<}),$ (13)

with

$\displaystyle
D_{\theta,>}:=\\{x\in\mathbb{R}^{2}:x_{2}>0,e^{x_{1}}{x_{2}}^{-1/p}>\theta\\},$
(14)

and

$\displaystyle
D_{\theta,<}:=\\{x\in\mathbb{R}^{2}:x_{2}>0,e^{x_{1}}{x_{2}}^{-1/p}<\theta\\}.$
(15)

######  Remark 2.4.

Note, that the points satisfying the infimum condition $F(x_{1},x_{2})=\theta$
in (11) are exactly those on the boundary
$\\{(x_{1},x_{2})\in\mathbb{R}^{2}:x_{2}>0,e^{x_{1}}{x_{2}}^{-1/p}=\theta\\}$
of $D_{\theta,>}$ and $D_{\theta,>}$ (which coincide). Hence, (11) shows that
the infimum of $\Lambda_{p}^{*}$ over this boundary is uniquely attained at
$\theta^{*}$.

As mentioned in the introduction, we will proceed to give an asymptotic
density approximation $h^{(n)}$ for $S^{(n)}$ in the following section, such
that for sufficiently large $n\in\mathbb{N}$ we can write the probabilities in
(13) as integrals of $h^{(n)}$ over $D_{\theta,>}$ and $D_{\theta,<}$. The
integral approximation result by Adriani and Baldi [3], which we use to
calculate that integral explicitly in the final section, will then show that
the values of the above integrals are heavily dependent of the geometric form
of the boundaries of $D_{\theta,>}$ and $D_{\theta,<}$. Thus, the direct
influence of the mapping $F(x_{1},x_{2}):=e^{x_{1}}{x_{2}}^{-1/p}$ on the LDP
through the contraction principle will have a more subtle form in the SLD
setting, as it influences the central results via the geometric properties of
its graph, seen as the local boundary of the sets $D_{\theta,>}$ and
$D_{\theta,<}$.

##  3\. Density Approximation

The sought-after density approximation for
$S^{(n)}=\frac{1}{n}\sum_{i=1}^{n}(\log|Y_{i}^{(n)}|,|Y_{i}^{(n)}|^{p})$ can
be derived by the original result of Borovkov and Rogozin [6, Theorem 1],
however we use a more convenient version of it from [3, Theorem 3.1]. For a
sum of i.i.d. random vectors with bounded common density it provides a local
density estimate for their empirical averages. We will not delve too deep into
the inner workings of the proof, however we will outline its core idea, so as
to argue why this result is still applicable to $S^{(n)}$, even though the
random vectors $(\log|Y_{i}^{(n)}|,|Y_{i}^{(n)}|^{p})$ clearly have no common
bounded density in $\mathbb{R}^{2}$, as their coordinates are highly
dependent. We start off by stating the result itself in the two-dimensional
setting, using the identities in [3, Equation (2.3)].

######  Proposition 3.1.

Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of $\mathbb{R}^{2}$-valued random
vectors. Assume that their common law $\mu$ has a bounded density with respect
to the Lebesgue measure and that their Laplace transforms are finite in a
neighbourhood of the origin. Then, for every for every $x$ in the effective
domain of the Legendre-Fenchel transform of the cumulant generating function
$\Lambda$, $S^{(n)}:=\frac{1}{n}\sum_{i=1}^{n}X_{n}$ has a density $h^{(n)}$
with

$h^{(n)}(x)={\frac{n}{2\pi}}\,{(\det\mathfrak{H}_{x})}^{-1/2}\,e^{-n\,\Lambda^{*}(x)}(1+o(1)),$

where $\mathfrak{H}_{x}:=\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))$ as in (3).

This is proven using the so called saddle point method (see e.g. Jensen [12]
for details), which provides a local density of the empirical average via the
Fourier transform of the involved i.i.d. random vectors, which are often
easier to obtain than their actual densities. Using the Fourier inversion
theorem, the (unknown) density of the empirical average is written as a
complex integral of the Fourier transform. Cauchy’s theorem then makes is
possible to transform the path of integration in such a way that it passes
through a complex saddle point, such that the mass of the integral heavily
concentrates in $n\in\mathbb{N}$ around this saddle point. Then, well-
established integral approximation techniques can be used with high accuracy,
to write the density explicitly.

The above process of course requires the involved Fourier transform to be
integrable in order to apply the Fourier inversion theorem, which is ensured
in [3] by the i.i.d. random vectors having a bounded common density. It is,
however, remarked in [3, Remark 3.2], that any other argument to the same
effect could be applied as well. In the context of this paper, the Fourier
transform of the random vectors
$V^{(n)}_{i}:=(\log|Y_{i}^{(n)}|,|Y_{i}^{(n)}|^{p})$ can be obtained via the
underlying distribution $\mathbf{N}_{p}$ of the random variables
$Y_{i}^{(n)}$. One can then deduce the integrability of the Fourier transform
of their empirical average for sufficiently large $n\in\mathbb{N}$, i.e. for
$n>n_{0}$, which is dependent on $x$ and $p$, by using the properties of the
density $f_{p}$ of $\mathbf{N}_{p}$ and applying the Hausdorff-Young
inequality. Since this was done in detail by Liao and Ramanan in [16, Lemma
6.1] for a sequence of random vectors that is very similar to ours, the proof
will be completely analogue and we refrain from reiterating it here. Thus,
despite the lack of a bounded density for the $V^{(n)}_{i}$, Proposition 3.1
can still be applied (with the added condition of $n\in\mathbb{N}$ being
sufficiently large), and yields the following proposition.

######  Proposition 3.2.

Let $p\in[1,\infty)$ and $n\in\mathbb{N}$. For
$S^{(n)}=\frac{1}{n}\sum_{i=1}^{n}V^{(n)}_{i}$ with
$V^{(n)}_{i}=(\log{|Y^{(n)}_{i}|},|Y^{(n)}_{i}|^{p}),$
$\,Y_{i}^{(n)}\sim\mathbf{N}_{p}$ i.i.d., $x\in\mathcal{J}_{p}$, and $n$
sufficiently large, it holds that the distribution of $S^{(n)}$ has Lebesgue
density

$h^{(n)}(x)=\frac{n}{2\pi}\,{(\det\mathfrak{H}_{x})}^{-1/2}\,e^{-n\,\Lambda_{p}^{*}(x)}\,(1+o(1)),$

with $\mathfrak{H}_{x}$ as in (3).

##  4\. Proof of Main Result

Assuming the set-up of Theorem 1.1 and combining the probabilistic
representation results in (8) and (13) with the local density approximation in
Proposition 3.2, we get that

$\displaystyle\mathbb{P}(\mathcal{R}_{n}>\theta)=\mathbb{P}(S^{(n)}\in
D_{\theta,>})$ $\displaystyle=$
$\displaystyle\int_{D_{\theta,>}}h^{(n)}(x)\,\textup{d}x$ (16)
$\displaystyle=$
$\displaystyle\frac{n}{2\pi}\,\int_{D_{\theta,>}}{(\det\mathfrak{H}_{x})}^{-1/2}\,e^{-n\,\Lambda_{p}^{*}(x)}\,\textup{d}x\,(1+o(1)),$

and

$\displaystyle\mathbb{P}(\mathcal{R}_{n}<\theta)=\mathbb{P}(S^{(n)}\in
D_{\theta,<})$ $\displaystyle=$
$\displaystyle\int_{D_{\theta,>}}h^{(n)}(x)\,\textup{d}x$ (17)
$\displaystyle=$
$\displaystyle\frac{n}{2\pi}\,\int_{D_{\theta,<}}{(\det\mathfrak{H}_{x})}^{-1/2}\,e^{-n\,\Lambda_{p}^{*}(x)}\,\textup{d}x\,(1+o(1)),$

with $D_{\theta,>}$ and $D_{\theta,<}$ as in (14) and (15). The final step of
the proof of our main result now is to calculate the above integrals
explicitly. We will only do this in detail for the integral in (16), as the
calculation for the integral in (17) proceeds in a mostly analogue fashion,
and we will merely point out the specific differences at the end of the proof.
As in [3, 14, 16], the first step is to split up the integration area into a
neighbourhood around the point $\theta^{*}$, at which the exponent in the
integrand attains its infimum on the boundary of $D_{\theta,>}$, and its
complement. On this neighbourhood we then employ a geometric result for
Laplace integral approximations by Adriani and Baldi [3], and on the
complement we use the large deviation principle from Proposition 2.3 to show
the comparative negligibility of the corresponding integral. The result for
Laplace integral approximations is geometric in the sense that it approximates
the integral using the Weingarten maps of the $\theta^{*}$-level set of
$\Lambda_{p}^{*}$ and the boundary of $D_{\theta,>}$ within the chosen
neighbourhood, both seen as planar curves.

For a brief recapitulation of the Weingarten map, we refer to [3, Section 4]
or [14, Section 2.5], and to [11, 15] for a more comprehensive resource. We
will simply note that in two-dimensional space, the Weingarten map of a curve
at a given point is simply the absolute value of its curvature at this point.
Hence, we introduce the following derivative notation and recall formulas for
curvatures of two types of planar curves: For a map
$f:\mathbb{R}^{d}\to\mathbb{R}$ and some
$x^{*}\in\mathbb{R}^{d},d\in\mathbb{N},$ we use the multi-index notation

$f_{[i_{1},\ldots,i_{d}]}(x^{*}):=\displaystyle\frac{\partial^{i_{1}}}{\partial
x_{1}^{i_{1}}}\ldots\frac{\partial^{i_{d}}}{\partial
x_{d}^{i_{d}}}\,f(x)\Big{|}_{x=x^{*}},$ (18)

with $i_{1},\ldots,i_{d}\in\mathbb{N}$. The following lemma provides formulas
for the curvature of planar curves, specifically for implicit curves, that is,
curves given as the zero set of a function, and for curves that are the graph
of a function. Both follow from the curvature formula given by Goldman in [10,
Proposition 3.1].

######  Lemma 4.1.

  * i)

Let $F:\mathbb{R}^{2}\to\mathbb{R}$ be a twice differentiable function. For a
curve $\mathscr{C}:=\\{x\in\mathbb{R}^{2}:F(x)=0\\}$ given as the zero set of
$F$, and a point $p\in\mathscr{C}$, where $\nabla_{x}F(p)\neq 0$, it holds for
the curvature $K$ of $\mathscr{C}$ in $p$ that

$K(p)=\displaystyle\frac{{F_{[0,1]}}^{2}{F_{[2,0]}}-2{F_{[0,1]}}{F_{[1,0]}}{F_{[1,1]}}+{F_{[1,0]}}^{2}{F_{[0,2]}}}{{\left({F_{[1,0]}}^{2}+{F_{[0,1]}}^{2}\right)}^{3/2}},$

with $F_{[i,j]}=F_{[i,j]}(p)$ as in (18).

  * ii)

In case that $\mathscr{C}$ is the graph of a twice differentiable function
$f:\mathbb{R}\to\mathbb{R}$, i.e.
$\mathscr{C}=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:x_{2}=f(x_{1})\\}$, and
$p=(x,f(x))$, the above reduces to

$K(p)=\displaystyle\frac{f^{\prime\prime}(x)}{{\big{(}1+f^{\prime}(x)^{2}\big{)}}^{3/2}}.$

The last result we will present before the proof of our main result is the
aforementioned Laplace integral approximation via the Weingarten map of
Adriani and Baldi [3]. The following proposition is the reduction of [3,
Section 4] into a singular concise result, as in [16, Lemma 5.6], for $d=2$.
Note, that for a set $D\subset\mathbb{R}^{2}$, we write $\partial
D,\overline{D},D^{\circ},$ and $D^{c}$ for its boundary, closure, interior and
complement, respectively.

######  Proposition 4.2.

Let $D\subset\mathbb{R}^{2}$ be a bounded domain such that $\partial D$ is a
differentiable planar curve in $\mathbb{R}^{2}$. Furthermore, let
$g:\mathbb{R}^{2}\to\mathbb{R}$ be a differentiable function and
$\phi:D\to[0,\infty)$ a nonnegative function that is twice differentiable and
attains a unique infimum over $\overline{D}$ at $x^{*}\in\partial D$. Define
the curves

$\mathscr{C}_{D}=\partial D\qquad\text{ and
}\qquad\mathscr{C}_{\phi}=\\{x\in\mathbb{R}^{2}:\phi(x)=\phi(x^{*})\\},$

and denote by $L_{D}$ and $L_{\phi}$ their respective Weingarten maps at
$x^{*}$. Then it holds that

$\displaystyle\int_{D}g(x)\,e^{-n\,\phi(x)}\,\textup{d}x$
$\displaystyle=\frac{{(2\pi)}^{1/2}\,\det(L_{\phi}^{-1}(L_{\phi}-L_{D}))^{-1/2}}{n^{3/2}\,\langle{\mathcal{H}_{x}\,\phi(x^{*})}^{-1}\,\nabla_{x}\phi(x^{*}),\nabla_{x}\phi(x^{*})\rangle^{1/2}}\,g(x^{*})\,e^{-n\,\phi(x^{*})}(1+o(1)).$

At this point, we have gathered the appropriate tools we need to proceed with
proving our main result.

###### Proof of Theorem 1.1.

We begin by proving the statement in Theorem 1.1 i). Let us assume the setting
therein and let $B_{\theta}$ be an open neighbourhood of $\theta^{*}$, small
enough such that $B_{\theta}\subset\mathcal{J}_{p}$. The fact that
$\theta^{*}\in\mathcal{J}_{p}$ follows from the fact that
$\Lambda_{p}^{*}(\theta^{*})=\mathcal{I}_{p}(\theta)<\infty$ for
$\theta\in(0,1)$, as seen in Proposition 2.3. Splitting up the reformulation
of our target probability in (16) into integrals of $h^{(n)}$ over
$B_{\theta}$ and $B_{\theta}^{c}$ yields

$\displaystyle\mathbb{P}(\mathcal{R}_{n}>\theta)$ $\displaystyle=$
$\displaystyle\int_{D_{\theta,>}\cap
B_{\theta}}h^{(n)}(x)\,\textup{d}x+\int_{D_{\theta,>}\cap
B^{c}_{\theta}}h^{(n)}(x)\,\textup{d}x.$ (19)

We begin by showing the comparative negligibility of the second integral term.
We know from Remark 2.4 that $\Lambda_{p}^{*}$ attains its unique infimum on
$\partial D_{\theta,>}$ at $\theta^{*}$. This property can be shown to hold
for the closure $\overline{D}_{\theta,>}$ as follows: assume
$t\in\mathbb{R}^{2}$ with $t\in D_{\theta,>}^{\circ}$, i.e.
$e^{t_{1}}t_{2}^{-1/p}>\theta$. We then consider
$\tilde{\theta}:=e^{t_{1}}t_{2}^{-1/p}$. If
$\tilde{\theta}^{*}\notin\mathcal{J}_{p}$, it trivially holds that
$\Lambda_{p}^{*}(\theta^{*})<\Lambda_{p}^{*}(t)=\infty$. Hence, assume that
$\tilde{\theta}^{*}\in\mathcal{J}_{p}$. It now follows that $t\in\partial
D_{\tilde{\theta},>}$, which yields that
$\Lambda_{p}^{*}(t)>\Lambda_{p}^{*}(\tilde{\theta}^{*})=\mathcal{I}_{p}(\tilde{\theta})$
by Remark 2.4. By the same arguments as in [3, p. 375], we know that the
Hessian of $\Lambda_{p}^{*}$ is strictly positive definite on its effective
domain $\mathcal{J}_{p}$, and therefore $\Lambda_{p}^{*}$ is strictly convex
on $\mathcal{J}_{p}$. From Proposition 2.3 we have that
$\Lambda_{p}^{*}({e^{m_{p}}}^{*})=\mathcal{I}_{p}(e^{m_{p}})=0$, thus
$\mathcal{I}_{p}$ is strictly increasing on $(e^{m_{p}},1)$, thus for
$\tilde{\theta}>\theta$ we have
$\Lambda_{p}^{*}(t)>\mathcal{I}_{p}(\tilde{\theta})>\mathcal{I}_{p}(\theta)=\Lambda_{p}^{*}(\theta^{*})$,
thereby proving that $\Lambda_{p}^{*}$ attains its unique infimum on
$\overline{D}_{\theta,>}$ at $\theta^{*}$. Therefore, it follows from
$\theta^{*}\notin B_{\theta}^{c}$ that there is an $\eta>0$, such that

$\inf_{t\in
D_{\theta}\cap\,B_{\theta}^{c}}\Lambda_{p}^{*}(t)>\Lambda_{p}^{*}(\theta^{*})+\eta.$

The LDP in Proposition 2.3 then implies that

$\limsup_{n\to\infty}\frac{1}{n}\log\mathbb{P}(S^{(n)}\in
D_{\theta}\cap\,B_{\theta}^{c})\leq-\inf_{y\,\in\,D_{\theta}\cap\,B_{\theta}^{c}}\Lambda_{p}^{*}(y)\leq-\Lambda_{p}^{*}(\theta^{*})-\eta,$

from which it follows that

$\mathbb{P}\left(S^{(n)}\in D_{\theta}\cap\,B_{\theta}^{c}\right)\leq
e^{-n\,\Lambda_{p}^{*}(\theta^{*})-n\,\eta}\,(1+o(1))=\frac{1}{e^{n\,\eta}}\,e^{-n\Lambda_{p}^{*}(\theta^{*})}(1+o(1)).$
(20)

Due to the leading exponential term $e^{-n\,\eta}$, the above will be
comparatively negligible compared to the other integral term

$\displaystyle\int_{D_{\theta,>}\cap B_{\theta}}h^{(n)}(x)\,\textup{d}x$
$\displaystyle=\frac{n}{2\pi}\,\int_{D_{\theta,>}\cap
B_{\theta}}{(\det\mathfrak{H}_{x})}^{-1/2}\,e^{-n\,\Lambda_{p}^{*}(x)}\,\textup{d}x\,(1+o(1)),$
(21)

which we will concretely calculate in the following. The clear course of
action for this will be to apply Proposition 4.2 to the integral in (4) with
$D=D_{\theta,>}\cap\,B_{\theta}\subset\mathbb{R}^{2}$, $x^{*}=\theta^{*}$,
$g(x):={(\det\mathfrak{H}_{x})}^{-1/2}$ and $\phi(x)=\Lambda_{p}^{*}(x)$.
However, we first need to check whether the conditions of Proposition 4.2
indeed hold. The area of integration is clearly bounded and since for
sufficiently small $B_{\theta}$ it follows from (14) that
$\partial(D_{\theta,>}\cap B_{\theta})$ around $\theta^{*}$ is a section of
the graph of the differentiable function $f(t_{1})=\theta^{-p}e^{pt_{1}}$, it
is indeed a differentiable planar curve. For the twofold differentiability of
$\Lambda_{p}^{*}$ we refer to the argument in the proof of the main result in
[14, p. 259] based on properties of the moment generating function and the
Legendre-Fenchel transform and the implicit function theorem, as it can be
applied in the same fashion here to show infinite differentiability of
$\Lambda_{p}^{*}$. This, in turn, also yields the infinite differentiability
of $\mathfrak{H}_{x}=\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))$, and hence
$g(x):={(\det\mathfrak{H}_{x})}^{-1/2}$ is differentiable. As
$\Lambda_{p}^{*}$ is a rate function for the LDP satisfied by $S^{(n)}$, it
follows by the standard properties of rate functions that it is non-negative.
Finally, as was shown in [13], it attains a unique infimum on
$\partial(D_{\theta,>}\cap B_{\theta})$ in $\theta^{*}$ (see (11) and Remark
2.4), which also holds for the entirety of $\overline{D_{\theta,>}\cap
B_{\theta}}$, as was shown above. Hence, we can apply Proposition 4.2 as
intended, which gives

$\displaystyle\displaystyle\int_{D_{\theta,>}\cap
B_{\theta}}h^{(n)}(x)\,\textup{d}x$
$\displaystyle=\displaystyle\frac{1}{\sqrt{2\pi
n}}\,\frac{\det(L_{\Lambda}^{-1}(L_{\Lambda}-L_{D}))^{-1/2}\,e^{-n\,\Lambda_{p}^{*}(\theta^{*})}}{\langle{\mathcal{H}_{x}\,\Lambda_{p}^{*}(\theta^{*})}^{-1}\,\nabla_{x}\,\Lambda_{p}^{*}(\theta^{*}),\nabla_{x}\,\Lambda_{p}^{*}(\theta^{*})\rangle^{1/2}\,(\det\mathfrak{H}_{\theta^{*}})^{1/2}}\,(1+o(1)),$
(22)

where $L_{\Lambda}$ and $L_{D}$ are the respective Weingarten maps of the
curves

$\mathscr{C}_{D}=\partial(D_{\theta,>}\cap\,B_{\theta})\qquad\text{ and
}\qquad\mathscr{C}_{\Lambda}=\\{x\in\mathbb{R}^{2}:\Lambda_{p}^{*}(x)=\Lambda_{p}^{*}(\theta^{*})\\}$

at $\theta^{*}$. We now need to resolve the different components in this
fraction. It was shown in [14, Lemma 21] that
$\nabla_{x}\Lambda_{p}^{*}(x)=\tau(x),$ and
$\mathcal{H}_{x}\Lambda_{p}^{*}(x)=\mathfrak{H}_{x}^{-1}$, which holds in our
setting by the very same arguments as presented therein. This allows rewriting
the term in the denominator in (4) as

$\displaystyle\Big{\langle}{\mathcal{H}_{x}\,\Lambda_{p}^{*}(\theta^{*})}^{-1}\,\nabla_{x}\,\Lambda_{p}^{*}(\theta^{*}),\nabla_{x}\,\Lambda_{p}^{*}(\theta^{*})\Big{\rangle}\det\mathfrak{H}_{\theta^{*}}\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\quad=\Big{\langle}\mathfrak{H}_{\theta^{*}}\,\tau(\theta^{*}),\tau(\theta^{*})\Big{\rangle}\det\mathfrak{H}_{\theta^{*}}=\xi(\theta)^{2}.$
(23)

In the following we shall give the Weingarten maps of the curves
$\mathscr{C}_{D}$ and $\mathscr{C}_{\Lambda}$ explicitly and see that
$\det(L_{\Lambda}^{-1}(L_{\Lambda}-L_{D}))=1-{L_{D}}/{L_{\Lambda}}=\kappa(\theta)^{2}$
(the determinant falling away due to the Weingarten maps being one-
dimensional). As discussed at the beginning of the section, the Weingarten map
of a planar curve at a given point reduces to the absolute value of its
curvature at that point, for which we have given concrete formulas in Lemma
4.1. As $\mathscr{C}_{D}=\partial(D_{\theta,>}\cap\,B_{\theta})$ around
$\theta^{*}$ is a segment of the graph of $f(t_{1})=\theta^{-p}e^{pt_{1}}$, we
get from Lemma 4.1 ii) that

$L_{D}=\frac{|f^{\prime\prime}(\theta)|}{(1+f^{\prime}(\theta)^{2})^{3/2}}=\frac{p^{2}e^{p\theta}\theta^{-p}}{\left(1+p^{2}e^{2p\theta}\theta^{-2p}\right)^{3/2}}.$
(24)

The curve $\mathscr{C}_{\Lambda}$ can be written as the zero set of the
function $F(x):=\Lambda_{p}^{*}(x)-\Lambda_{p}^{*}(\theta^{*})$, and its
derivatives $F_{[i,j]}$ at $\theta^{*}$ as in Lemma 4.1 i) are known from the
identities $\nabla_{x}\Lambda_{p}^{*}(x)=\tau(x)$ and
$\mathcal{H}_{x}\Lambda_{p}^{*}(x)={\mathfrak{H}_{x}}^{-1}$ from [14, Lemma
21]. (Note, that for $\theta=e^{m_{p}}$ we have that $\mathscr{C}_{\Lambda}$
is the zero set of $F(x)=\Lambda_{p}^{*}(x)$, since
$\Lambda_{p}^{*}({e^{m_{p}}}^{*})=0$. By (12) it follows that $\tau(x)=0$ only
if $x={e^{m_{p}}}^{*}$. Hence, the zero set of $F(x)=\Lambda_{p}^{*}(x)$ is
solely ${e^{m_{p}}}^{*}$, which is not a differentiable curve, and hence is
not accessible by these geometric methods). It thus follows that

$L_{\Lambda}=\displaystyle\frac{\left|\tau(\theta^{*})_{2}^{2}\left(\mathfrak{H}_{\theta^{*}}^{-1}\right)_{11}-2\tau(\theta^{*})_{1}\tau(\theta^{*})_{2}\left(\mathfrak{H}_{\theta^{*}}^{-1}\right)_{12}+\tau(\theta^{*})_{1}^{2}\left(\mathfrak{H}_{\theta^{*}}^{-1}\right)_{22}\right|}{{\big{(}\tau(\theta^{*})_{1}^{2}+\tau(\theta^{*})_{2}^{2}\big{)}}^{3/2}}.$

This, together with (24), now yields that
$1-{L_{D}}/{L_{\Lambda}}=\kappa(\theta)^{2}$, which combined with (11) and (4)
gives

$\displaystyle\displaystyle\int_{D_{\theta,>}\cap\,B_{\theta}}h^{(n)}(x)\,\textup{d}x$
$\displaystyle=\displaystyle\frac{1}{\sqrt{2\pi
n}\,\xi(\theta)\,\kappa(\theta)}e^{-n\mathcal{I}_{p}(\theta)}\,(1+o(1)).$ (25)

Comparing (25) with the upper bound of the integral outside of $B_{\theta}$ in
(20), we can see that the integral over $B_{\theta}^{c}$ is negligible as it
is of order $o(1)$. Thus, combining (19), (20) and (25) finishes the proof of
Theorem 1.1 i).

The proof of Theorem 1.1 ii) is almost completely the same regarding
probabilistic representation, local density estimation and integral
approximation, as hardly any of the steps therein use the fact that we are
working on $D_{\theta,>}$ for $\theta\in(e^{m_{p}},1)$ instead of
$D_{\theta,<}$ for $\theta\in(0,e^{m_{p}})$, but rather consider a
neighbourhood of $\partial D_{\theta,>}$ around $\theta^{*}$, which coincides
with that same neighbourhood of $\partial D_{\theta,<}$ around $\theta^{*}$,
and are therefore the same in both settings. The only notable difference is
that one shows the fact that $\theta^{*}$ minimizes $\Lambda_{p}^{*}$ not only
on $\partial D_{\theta,<}$, as in (11), but also on $\overline{D}_{\theta,<}$,
by using the fact that $\Lambda_{p}^{*}$ is strictly decreasing on
$(0,e^{m_{p}})$ instead of it being strictly increasing on
$\theta\in(e^{m_{p}},1)$. Beyond this, the proof is the same as for Theorem
1.1 i) and is hence omitted here. ∎

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