# Sharp concentration phenomena in high-dimensional Orlicz balls

Lorenz Frühwirth and Joscha Prochno

###### Abstract

In this article, we present a precise deviation formula for the intersection
of two Orlicz balls generated by Orlicz functions $V$ and $W$. Additionally,
we establish a (quantitative) central limit theorem in the critical case and a
strong law of large numbers for the "$W$-norm" of the uniform distribution on
$\mathbb{B}^{(n,V)}$. Our techniques also enable us to derive a precise
formula for the thin-shell concentration of uniformly distributed random
vectors in high-dimensional Orlicz balls. In our approach we establish an
Edgeworth-expansion using methods from harmonic analysis together with an
exponential change of measure argument.

##  1 Introduction & Main results

Let $n\in\operatorname{\mathbb{N}},1\leq p<\infty$ and let
$\mathbb{D}^{(n,p)}$ be the volume normalized $\ell_{p}$-ball in
$\operatorname{\mathbb{R}}^{n}$. In [46] Schechtman and Schmuckenschläger
studied the asymptotic behavior of the intersection of $\mathbb{D}^{(n,p)}$
with an appropriately scaled $\ell_{q}$-ball and found the threshold behavior

$\operatorname{vol}_{n}\left(\mathbb{D}^{(n,p)}\cap
t\mathbb{D}^{(n,q)}\right)\xrightarrow{n\rightarrow\infty}\begin{cases}0,&t<A_{p,q}^{-1}\\\
1,&t>A_{p,q}^{-1},\end{cases}$ (1)

where $A_{p,q}\in(0,\infty)$ is an explicit constant. Their proof mainly
relied on a probabilistic representation of the uniform distribution on
$\ell_{p}^{n}$-balls obtained by Schechtman and Zinn [47] and independently by
Rachev and Rüschendorf [44]). More precisely, for
$X^{(n,p)}\sim\mathbb{D}^{(n,p)}$, we can write

$X^{(n,p)}\stackrel{{\scriptstyle
d}}{{=}}U^{1/n}\frac{\left(Y_{1},\ldots,Y_{n}\right)}{||\left(Y_{1},\ldots,Y_{n}\right)||_{p}},$
(2)

where $U\sim\operatorname{\mbox{Unif}}([0,1])$ and
$(Y_{i})_{i\in\operatorname{\mathbb{N}}}$ are independent and identically
distributed (iid) with Lebesgue density

$\operatorname{\mathbb{R}}\ni
x\mapsto\frac{1}{2p^{1/p}\Gamma(1+1/p)}e^{-|x|^{p}/p}.$

It is then possible to represent the volume in (1) as the probability that a
random vector, which is uniformly distributed in the (normalized)
$\ell_{p}^{n}$-ball, has $\ell_{q}$-norm smaller than $t$. The probabilistic
representation of $X^{(n,p)}$ allows us to work with independent random
variables and thus the limit in (1) is essentially an application of the
strong law of large numbers. The threshold case $t=A_{p,q}$ first remained an
open problem before Schmuckenschläger proved a central limit theorem for the
$\ell_{q}$-norm of a point chosen randomly in an $\ell_{p}^{n}$-ball [49]. The
existence of a probabilistic representation as in (2) gave rise to the rich
and flourishing theory of high-dimensional $\ell_{p}$-geometry, which was
intensively studied over the past 15 years (we refer the interested reader to
the surveys [41, 43]). One aim of the present paper is to generalize the
Schechtman–Schmuckenschläger result from (1) in a different direction. All
mentioned results above heavily use the representation in (2), which is
limited to $\ell_{p}$-balls. Here we want to study concentration phenomena
like (1) for the intersection of so-called Orlicz balls. Typically, taking a
symmetric and convex function
$V:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ with $V(x)=0$ if and only
if $x=0$ and some $R\in(0,\infty)$, the associated Orlicz ball with "radius"
$R$ is defined as

$\mathbb{B}^{(n,V)}_{R}:=\left\\{x\in\operatorname{\mathbb{R}}^{n}\,:\,\sum_{i=1}^{n}V(x_{i})\leq
nR\right\\};$ (3)

in this work we deal with a slightly more restrictive notion of an Orlicz
function (see Definition 2.1 and Remark 2.2). Orlicz spaces serve as natural
generalizations of $\ell_{p}$-spaces and are fundamental elements within the
class of symmetric Banach sequence spaces. They have been intensively studied
in the functional analysis literature and we direct the interested reader to
[20, 26, 35, 36, 50, 42, 45] and the references cited therein. In [22], using
the maximum entropy principle from statistical mechanics and elements from
(sharp) large deviations theory, Kabluchko and Prochno were able to bypass the
non-existence of a classical Schechtman–Schmuckenschläger-type probabilistic
representation and computed the asymptotic volume of high-dimensional Orlicz
balls; independently similar problems were considered by Barthe and Wolff in
[8] with a view towards confirming the Kannan–Lovász–Simonovits (KLS) spectral
gap conjecture. Although not stated explicitly it follows from the proof of
[22, Theorem B] that the convergence rate is exponential. In our Theorem A we
are able to present a precise version of [22, Theorem B], which also includes
lower-order terms and could not be obtained by the approach developed in [22].
The study of moderate and large deviations principles in the realm of
asymptotic geometric analysis has been relatively recent, pioneered by
Gantert, Kim, and Ramanan in [18], and further developed by Prochno,
Kabluchko, and Thäle in [25]. In their recent work [4], Alonso-Gutiérrez,
Prochno, and Thäle uncovered a connection between the investigation of
moderate and large deviations for isotropic log-concave random vectors and the
famous KLS conjecture. Unlike central limit theorems, (sharp) large deviation
principles are highly sensitive to the distribution of the underlying random
variables. Although large deviations principles (LDPs), unlike concentration
results or large deviation upper bounds, precisely identify the asymptotic
exponential decay rate and allow for the identification of conditional limit
laws (see [16, 29]), they have a notable drawback. In general, LDPs only
provide approximate estimates of the probability that a sequence of random
variables lies in an atypical region. Specifically, they characterize the
limit of the logarithms of the deviation probabilities as the dimension $n$
tends to infinity. Sharp LDPs, on the other hand, not only provide the precise
exponential decay but also include additional subexponential factors, offering
more accurate estimates and are more in the spirit of asymptotic geometric
analysis.

While numerous articles investigate the asymptotic behavior of high-
dimensional $\ell_{p}$-balls and spheres (see, e.g., [5, 17, 18, 23, 24, 25,
27, 29]), the study becomes more nuanced and delicate when extending to high-
dimensional Orlicz balls. This can be seen in the work of Kabluchko and
Prochno [22] and the complexity stems from the absence of a probabilistic
representation for the uniform distribution on $\mathbb{B}^{(n,V)}_{R}$ and
because general Orlicz functions are not homogenous. The former problem can be
resolved by employing suitable Gibbs measures $\mu_{\alpha}$, distributed
according to the Lebesgue density

$\frac{d\mu_{\alpha}}{dx}(x)=\frac{e^{\alpha V(x)}}{Z_{\alpha}},\quad
x\in\mathbb{R},$

where, in statistical mechanics terminology, $\alpha\in(-\infty,0)$ is
referred to as the inverse critical temperature, and $Z_{\alpha}$ is the
partition function, i.e., the normalizing constant; this approach was
initiated in [22] and has led to successful breakthroughs in addressing
geometric questions within high-dimensional Orlicz spaces by probabilistic
means (see, e.g., [3, 16, 21, 28, 29]). Turning to sharp large deviations,
those can be traced back to Bahadur and Rao [7] and Petrov [40] (see also
[15]). Essentially, they used an exponential change of measure to identify the
exponential decay of the probability that the empirical mean of iid random
variables exceeds their common mean. The residual term can then be
approximated using a first-order Edgeworth expansion (see, e.g., [40, p.
293]). The proof of Theorem A adopts a similar approach, but encounters
certain challenges in establishing an Edgeworth expansion for the resulting
sum of random variables (see Section 3.2 and the discussion therein). After
identifying the precise asymptotic formula for the
Schechtman–Schmuckenschläger ratio (1) in the case of Orlicz balls, one may
raise questions about its behavior in the critical case; note that this
question could not be addressed by the methods in [22]. This inquiry is
addressed in our Theorem B, where we demonstrate that the normalized volume
intersection for high-dimensional Orlicz balls adheres to a quantitative
Central Limit Theorem with a speed of order $n^{-1/2}$, analogous to the
classical Berry–Esseen theorem.

Our techniques also allow to give a precise description of the so-called thin-
shell concentration of high-dimensional Orlicz balls (see Theorem C). Roughly
speaking, for small values of $\delta\in(0,\infty)$, we provide a precise
formula for

$\mathbb{P}\left[\left|\frac{\|X^{(n,V)}\|_{2}^{2}}{n}-m\right|\geq\delta\right],$
(4)

where $X^{(n,V)}$ follows a uniform distribution on an Orlicz ball generated
by the Orlicz function $V$. Estimating probabilities of the form (4) has
attracted considerable attention since the works of Sudakov [51], Diaconis and
Freedman [14], von Weizsäcker [52], and Antilla, Ball, and Perissinaki [6]. We
also highlight the breakthrough result by Klartag [30], who proved that the
marginals of an isotropic random vector are close to a Gaussian distribution.
In [6], it was established that any isotropic (with mean $0$ and identity
covariance matrix) random vector $X^{(n)}\in\operatorname{\mathbb{R}}^{n}$ has
(approximately) Gaussian marginals if $\|X^{(n)}\|_{2}/\sqrt{n}$ concentrates
around $1$. In other words, $||X^{(n)}||_{2}$ concentrates in a thin-shell
with radius $\sqrt{n}$. This principle has led to the thin-shell width
conjecture, which proposes that there exists an absolute constant
$C\in(0,\infty)$ such that for every $n\in\operatorname{\mathbb{N}}$ and every
isotropic random vector $X^{(n)}\in\operatorname{\mathbb{R}}^{n}$ one has
$\mathbb{E}\left[\|X^{(n)}\|_{2}-\sqrt{n}\right]^{2}\leq C$. This is
equivalent to the famous variance conjecture, first formally proposed in [10]
(see also [1]). It asserts the existence of an absolute constant
$C\in(0,\infty)$ such that for any isotropic random vector
$X^{(n)}\in\operatorname{\mathbb{R}}^{n}$, it holds that
$\mathbb{V}\left[\|X^{(n)}\|_{2}^{2}\right]\leq Cn$. The variance conjecture
has been successfully approached for various important examples of isotropic
convex bodies (see, e.g., [53], [34] and the references cited therein). The
variance conjecture is a special case of the renowned KLS conjecture, which
roughly says that, up to a universal constant $C\in(0,\infty)$ independent of
the dimension $n$, the best way to divide a convex body into two pieces of
equal mass in order to minimize their interface is a hyperplane bisection. For
a long time the best known bound was obtained by Bourgain [11, 12] where it is
proven that $C=O\left(n^{1/4}\log n\right)$. A few years ago Chen [13]
accomplished a breakthrough by obtaining a subpolynomial bound for the KLS
constant. This work led to great further improvements (e.g., by Klartag and
Lehec [32]) with the currently best known estimate for the KLS constant of
$O(\sqrt{\log n})$ obtained by Klartag in [31] (see also the recent survey
[33] for a more comprehensive discussion on the KLS conjecture and related
topics).

Returning to thin-shell concentration estimates, we mention that in 2018, Lee
and Vempala [37] proved that for $t\in(0,\infty)$,

$\mathbb{P}\left[\left|\frac{\|X^{(n)}\|_{2}}{\mathbb{E}\left[\|X^{(n)}\|_{2}\right]}-1\right|\geq
t\right]\leq
e^{-c\min\left\\{t,t^{2}\right\\}\sqrt{n}}\quad\text{and}\quad\mathbb{E}\left[\|X^{(n)}\|_{2}-\sqrt{n}\right]^{2}\leq
C\sqrt{n};$

another estimate of a similar order was obtained by Guédon and Milman [19]
(see also [39], [48], and [2] for more precise estimates in case of
$\ell_{p}$-balls). More recently, Prochno and Alonso-Gutierrez [3], using a
moderate deviations perspective, were able to provide precise estimates for
probabilities of the form

$\mathbb{P}\left[\left|\frac{\|X^{(n)}\|_{2}^{2}}{nL_{Z}^{2}}-1\right|\geq
t_{n}\right],$

where $X^{(n)}$ is uniformly distributed in a high-dimensional Orlicz ball
$\mathbb{B}^{(n,V)}_{R}$, $L_{Z}^{2}$ is the asymptotic value of the isotropic
constant (see also [3, Theorem A]) and
$(t_{n})_{n\in\operatorname{\mathbb{N}}}$ is a sequence that grows to infinity
slower than $\sqrt{n}$. Their approach requires the Orlicz function $V$ to
grow sufficiently fast; more precisely, they assumed that

$\lim_{x\rightarrow\infty}\frac{V(x)}{x^{2}}=\infty.$

In our Theorems A and C, we are able to weaken this assumption and we only
require $V$ to grow at least as fast as $x\mapsto x^{2}$ (see also Assumption
A).

###  1.1 Main results

Before stating our main results, Theorem A, B and C, we introduce a few
quantities. An Orlicz function is a symmetric and convex function
$V:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ with $V(x)=0$ if and only
if $x=0$, which is essentially $\mathcal{C}_{2}$ (see Section 2.2 for a
precise definition). The associated Orlicz ball $\mathbb{B}_{R}^{(n,V)}$ with
radius $R\in(0,\infty)$ in $\operatorname{\mathbb{R}}^{n}$ is then defined as
in (3). For $n\in\operatorname{\mathbb{N}}$ and $R\in(0,\infty)$, let
$X^{(n,V)}\sim\operatorname{Unif\left(\mathbb{B}_{R}^{(n,V)}\right)}$. Then,
as shown in Theorem B, for any (linearly independent) Orlicz functions $V$ and
$W$, the almost sure limit

$m:=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}W\left(X_{i}^{(n,V)}\right)\in(0,\infty)$
(5)

exists.

Let $V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be two Orlicz
functions and, for $(\alpha,\beta)\in\operatorname{\mathbb{R}}^{2}$, we define
a measure $\mu_{\alpha,\beta}$ via the Lebesgue density

$\frac{d\mu_{\alpha,\beta}(x)}{dx}=\frac{e^{\alpha V(x)+\beta
W(x)}}{Z_{\alpha,\beta}},\quad x\in\operatorname{\mathbb{R}},$

provided $Z_{\alpha,\beta}:=\int_{\operatorname{\mathbb{R}}}e^{\alpha
V(x)+\beta W(x)}dx<\infty$. Gibbs measures of this or a similar form have
proven to be very useful when dealing with the asymptotic behavior of high-
dimensional Orlicz balls (for more background on the maximum entropy principle
for Orlicz spaces, we refer to Section 1.2 in [22]). Moreover, we define
$\varphi:\operatorname{\mathbb{R}}^{2}\rightarrow(-\infty,\infty]$ with

$\varphi(\alpha,\beta):=\log Z_{\alpha,\beta}.$

Let
$\operatorname{\mathbb{I}}:\operatorname{\mathbb{R}}^{2}\rightarrow[0,\infty]$
be the Legendre transform of $\varphi$, i.e., for
$(x,y)\in\operatorname{\mathbb{R}}^{2}$, we set

$\operatorname{\mathbb{I}}(x,y):=\sup_{(\alpha,\beta)\in\operatorname{\mathbb{R}}^{2}}\left[\alpha
x+\beta y-\varphi(\alpha,\beta)\right],$ (6)

which is well defined since $\varphi$ is convex.

######  Assumption A.

Let $V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be two Orlicz
functions. We say $V$ and $W$ satisfy Assumption A, if

$\liminf_{x\rightarrow\infty}\frac{V(x)}{W(x)}>0$

Assumption A essentially means that $W$ grows at most as fast as $V$.

######  Assumption B.

Let $V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be two Orlicz
functions. We say $V$ and $W$ satisfy Assumption B, if

$\lim_{x\rightarrow\infty}\frac{V(x)}{W(x)}=\infty.$

Assumption B requires $V$ to grow much faster than $W$ and we note that
Assumption B implies A. We highlight that similar conditions to Assumption B
were already imposed in, e.g., [3, 29].

######  Remark 1.1.

For example, Assumption B is satisfied for $V(x)=|x|^{p}$ and $W(x)=|x|^{q}$
if $p>q\geq 1$. The Orlicz functions $V(x):=x^{4}+x^{2}$ and $W(x):=x^{4}$
satisfy Assumption A, but not Assumption B (see also Remark 3.4, where we
demonstrate the necessity of distinguishing between Assumptions A and B!).

In the theory of large deviations, functions as $\operatorname{\mathbb{I}}$
from (6) are usually referred to as good rate functions (GRF). This means that
all level sets of $\operatorname{\mathbb{I}}$ are compact.

######  Theorem A.

$(i)$ Let $R\in(0,\infty)$ and
$V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be linearly independent
Orlicz functions that satisfy Assumption A. Then there exists an
$\varepsilon\in(0,\infty)$ such that for all $t\in(m,m+\varepsilon)$, we have

$\frac{\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\cap\mathbb{B}_{t}^{(n,W)}\right)}{\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\right)}=1-\frac{1}{\sqrt{2\pi
n}\sigma}e^{-n\operatorname{\mathbb{I}}(R,t)}(1+o(1)),$ (7)

where $m\in(0,\infty)$ is the constant from (5), $\operatorname{\mathbb{I}}$
is the GRF from (6) and $\sigma=\sigma(V,W,R,t)$ is an explicit positive
quantity.

$(ii)$ If $V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ satisfy
Assumption B, then (7) holds for all $t\in(m,W(V^{-1}(R))$.

######  Remark 1.2.

We observe that in the case of $W(x):=|x|^{p}$ for some $p\geq 1$, we can
express

$\displaystyle\mathbb{P}\left[\left|\left|X^{(n,V)}\right|\right|_{p}^{p}\geq
t\right]$
$\displaystyle=1-\frac{\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\cap\mathbb{B}_{t}^{(n,W)}\right)}{\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\right)}.$

Hence, Theorem A generalizes [27, Theorem 11] and provides a refinement of
[22, Theorem B]. In several related works (see, e.g.,[27, 38]) the admissible
values of $t$ are non-explicit. In Theorem A $(ii)$ we can identify the
explicit interval $(m,W(V^{-1}(R)))$ such that the sharp asymptotic formula
given in (7) holds for all values of $t$ in that interval. We consider this to
be a non-trivial feature of our result (see the proof of Lemma 3.2).

As in case of $\ell_{p}$-balls, we see that the "$W$-norm" of a uniform
distribution on a $V$-Orlicz ball $\mathbb{B}^{(n,V)}_{R}$ concentrates around
a critical value $m\in(0,\infty)$. The following theorem shows that this
concentration happens almost surely and, under a suitable normalization, we
obtain a Berry–Esseen-type result, i.e., a quantitative central limit theorem.
We denote the Kolmogoroff distance by $d_{\text{Kol}}$ (see Equation (11) for
a formal definition).

######  Theorem B.

(Strong law of large numbers and CLT) Let
$V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be linearly independent
Orlicz functions. For $n\in\operatorname{\mathbb{N}}$ and $R\in(0,\infty)$,
let
$\operatorname{X^{(n,V)}}\sim\operatorname{Unif\left(\mathbb{B}_{R}^{(n,V)}\right)}$,
then the limit

$m=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}W\left(X_{i}^{(n,V)}\right)$
(8)

exists almost surely with $m=\mathbb{E}_{\mu_{\alpha^{*},0}}[W(X)]$, where
$\alpha^{*}\in(0,\infty)$ is chosen such that
$\mathbb{E}_{\mu_{\alpha^{*},0}}[V(X)]=R$. Moreover, we can even give a
quantitative version of the limit above: there exists a constant
$c\in(0,\infty)$ such that

$d_{\text{Kol}}\left(\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[W\left(X_{i}^{(n,V)}\right)-m\right],Z\right)\leq\frac{c}{\sqrt{n}},\qquad
n\in\operatorname{\mathbb{N}},$ (9)

where $Z\sim\mathcal{N}(0,\sigma_{*}^{2})$ and $\sigma_{*}^{2}\in(0,\infty)$
is a different quantity as in Theorem C.

Moreover, the result remains valid if we choose $V(x)=|x|^{p}$ and
$W(x)=|x|^{q}$ for some $p,q\in(0,1)$ with $p\neq q$.

######  Remark 1.3.

It’s worth noting that Theorem B holds true for functions $V(x)=|x|^{p}$ and
$W(x)=|x|^{q}$, where $p,q\in(0,1)$. Remarkably, these functions are neither
Orlicz functions nor convex. We emphasize that a key component of the proof of
Theorem B is the establishment of an Edgeworth expansion for a specific
sequence of random vectors (see Section 3.2). We anticipate that the
techniques developed in Section 3.2 could apply to a broader class of
functions beyond those considered in Theorem B, which could be of independent
interest.

As a consequence of our CLT, we also obtain the following Corollary answering
a question in [22, p. 4].

######  Corollary 1.4.

Let $V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be two linearly
independent Orlicz functions. Then, for any radius $R\in(0,\infty)$, we have

$\frac{\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\cap\mathbb{B}_{m}^{(n,W)}\right)}{\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\right)}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}\frac{1}{2},$

where $m=m(R)\in(0,\infty)$ is the quantity from (5).

Another valuable application of our techniques is the following thin-shell
concentration result, which complements [3, Theorem B].

######  Theorem C.

(Sharp thin-shell concentration) Let $R\in(0,\infty)$, $W(x):=x^{2}$ and
$V:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be an Orlicz function such
that Assumption A is satisfied. Then, there exists an
$\varepsilon\in(0,\infty)$ such that for all $\delta\in(0,\varepsilon)$, we
have

$\mathbb{P}\left[\left|\frac{||X^{(n,V)}||_{2}^{2}}{n}-m\right|>\delta\right]=\frac{1}{\sqrt{2\pi
n}\tilde{\sigma}}e^{-n\min\\{\mathbb{I}(R,m+\delta),\mathbb{I}(R,m-\delta)\\}}(1+o(1))$
(10)

where
$\operatorname{X^{(n,V)}}\sim\operatorname{Unif\left(\mathbb{B}_{R}^{(n,V)}\right)}$
and $\operatorname{\mathbb{I}}$ is the GRF from (6) and $m\in(0,\infty)$ is
the almost sure limit from Theorem B (see also Equation 8). The quantity
$\tilde{\sigma}\in(0,\infty)$ depends on $V,R$, and $\delta$.

######  Remark 1.5.

In fact, we will prove a more general result as it is stated in Theorem C,
where the $2$-norm is replaced by the "$W$-norm" with an Orlicz function $W$
satisfying Assumption A.

In particular, we note that Assumption A is weaker than the superquadratic
assumption required in [3, Theorem B].

##  2 Notation and Preliminaries

###  2.1 Notation

Let $X$ be a real-valued random variable and $\mu$ be some distribution. If
$X$ is distributed according to $\mu$, we write $X\sim\mu$. We denote the
Dirac-measure in some point $x\in\operatorname{\mathbb{R}}$ by $\delta_{x}$.
The mean of $X$ is denoted by $\mathbb{E}\left[X\right]$ and for the variance
of $X$ we write $\mathbb{V}\left[X\right]$, provided the respective quantities
exist. Let $f:\operatorname{\mathbb{R}}\rightarrow\operatorname{\mathbb{R}}$
be some measurable function and $\mu$ some distribution; whenever we write

$\mathbb{E}_{\mu}\left[f(X)\right]\quad\text{or}\quad\mathbb{V}_{\mu}\left[f(X)\right],$

we indicate that $X\sim\mu$. Let $k\in\operatorname{\mathbb{N}}$ and
$\Sigma\in\operatorname{\mathbb{R}}^{k\times k}$ be a positive definite,
symmetric matrix and $v\in\operatorname{\mathbb{R}}^{k}$ be some vector. The
(multivariate) normal distribution with mean $v$ and covariance matrix
$\Sigma$ is then denoted by $\mathcal{N}\left(\mu,\Sigma\right)$. Throughout
this article we will only see the cases $k=1$ or $k=2$. Let $X$ and $Y$ be
random variables. The Kolmogoroff distance between $X$ and $Y$ is defined as

$d_{\text{Kol}}(X,Y):=\sup_{t\in\operatorname{\mathbb{R}}}\left|\mathbb{P}\left[X\leq
t\right]-\mathbb{P}\left[Y\leq t\right]\right|.$ (11)

Let $I\subseteq\operatorname{\mathbb{R}}$ be an interval with endpoints $a,b$.
In Section 3.2 we will frequently use the notation $|I|=b-a$ to indicate the
length of $I$. Let $A\subseteq\operatorname{\mathbb{R}}^{d}$ be a set, where,
depending on the context, we will use both notations $\mathbb{1}_{A}(x)$ and
$\mathbb{1}_{\left[x\in A\right]}$ for the indicator function over the set
$A$. Let $U\subseteq\operatorname{\mathbb{R}}^{2}$ be an open non-empty set
and $\varphi:U\rightarrow\operatorname{\mathbb{R}}$ be two times continuously
differentiable on $U$. We then write $\varphi\in\mathcal{C}^{2}(U)$ or if
$U=\operatorname{\mathbb{R}}^{2}$, we just write $\varphi\in\mathcal{C}^{2}$.
For $(x,y)\in U$, the gradient of $\varphi$ in $(x,y)$ is defined as

$\nabla\varphi(x,y):=\left(\begin{array}[]{c}\frac{\partial\varphi}{\partial
x}(x,y)\\\ \frac{\partial\varphi}{\partial y}(x,y)\end{array}\right).$

The Hessian of $\varphi$ in some point $(x,y)\in U$ is denoted by

$\displaystyle
H_{\varphi}(x,y):=\left(\begin{matrix}\frac{\partial^{2}\varphi}{\partial
x^{2}}(x,y)&\frac{\partial^{2}\varphi}{\partial x\partial y}(x,y)\\\
\frac{\partial\varphi}{\partial y\partial
x}(x,y)&\frac{\partial^{2}\varphi}{\partial y^{2}}(x,y)\end{matrix}\right).$

Let $(a_{n})_{n\in\operatorname{\mathbb{N}}}$ be a sequence of real numbers
and $(b_{n})_{n\in\operatorname{\mathbb{N}}}$ a sequence of monotone
decreasing and positive numbers. We write $a_{n}=o(b_{n})$, if, for all
$\varepsilon\in(0,\infty)$, there exists an $N\in\operatorname{\mathbb{N}}$
such that for all $n\geq N$, it holds that

$|a_{n}|\leq\varepsilon b_{n}.$

We write $a_{n}=O(b_{n})$, if there exists a constant $C\in(0,\infty)$ such
that, for all $n\in\operatorname{\mathbb{N}}$

$|a_{n}|\leq Cb_{n}.$

If all $a_{n}$ are non-negative, we also use the Vinogradov notation $a_{n}\ll
b_{n}$ if $a_{n}=O(b_{n})$. Let $(X_{n})_{n\in\operatorname{\mathbb{N}}}$ be a
sequence of real-valued random variables.

###  2.2 Orlicz functions & Gibbs measures

######  Definition 2.1.

A symmetric, convex function
$V:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ with $V(x)=0$ iff $x=0$ is
called an Orlicz function if $V$ is two times continously differentiable. In
fact, we only need that $V$ is $\mathcal{C}^{2}$ except of countably many
points that do not accumulate anywhere. We recall the definition of the
associated Orlicz ball $\mathbb{B}^{(n,V)}_{R}$ of radius $R\in(0,\infty)$
with

$\mathbb{B}_{R}^{(n,V)}=\left\\{x\in\operatorname{\mathbb{R}}^{n}:\sum_{i=1}^{n}V(x_{i})\leq
Rn\right\\}.$

######  Remark 2.2.

Our notion of an Orlicz function is slightly more restrictive than usual,
since we require that $V$ is $\mathcal{C}^{2}$ except of a discrete set of
points. Often (see, e.g., [3, 22]) $V$ is assumed to be just convex, which by
Alexandrov’s theorem, implies that $V$ is $\mathcal{C}^{2}$ almost everywhere.
The set of points where $V$ is not $\mathcal{C}^{2}$ could still be dense.
Here we exclude such examples. We believe that this assumption could be
overcome with a significant amount of technical work in Section 3.2, which we
have chosen not to undertake.

The uniform distribution of the intersection of two Orlicz balls
$\mathbb{B}_{R}^{(n,V)}$ and $\mathbb{B}_{R}^{(n,W)}$ is naturally related to
two-factor Gibbs measures $\mu_{\alpha,\beta}$ defined via the Lebesgue
density

$\frac{d\mu_{\alpha,\beta}}{dx}(x)=\frac{e^{\alpha V(x)+\beta
W(x)}}{Z_{\alpha,\beta}},\qquad x\in\operatorname{\mathbb{R}},$ (12)

where

$Z_{\alpha,\beta}=\int_{\operatorname{\mathbb{R}}}e^{\alpha V(x)+\beta
W(x)}dx$

is the normalizing constant and
$(\alpha,\beta)\in\operatorname{\mathbb{R}}^{2}$ are chosen in a way such that
$\mu_{\alpha,\beta}$ exists. For instance, if we choose $\alpha=-1$ and
$\beta=0$, we always have that $\mu_{\alpha,\beta}$ exists since $V$ is convex
as Orlicz function.

##  3 Proofs

###  3.1 Auxiliary Results

In this Section we work with Orlicz functions
$V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ and Gibbs measures
$\mu_{\alpha,\beta}$ with distribution given in (12), where
$(\alpha,\beta)\in\operatorname{\mathbb{R}}^{2}$ are chosen such that
$\mu_{\alpha,\beta}$ exists. We recall that
$Z_{\alpha,\beta}=\int_{\operatorname{\mathbb{R}}}e^{\alpha V(x)+\beta
W(x)}dx$, $\varphi(\alpha,\beta)=\log Z(\alpha,\beta)$ and define

$\displaystyle\mathcal{D}_{\varphi}:=\left\\{(\alpha,\beta)\in\operatorname{\mathbb{R}}^{2}:\varphi(\alpha,\beta)<\infty\right\\}.$

The following lemma gives (partial) characterisations of the set
$\mathcal{D}_{\varphi}$, depending on the relation of $V$ and $W$.

######  Lemma 3.1.

$(i)$ If $V$ and $W$ satisfy Assumption A, then

$\left\\{(\alpha,\beta)\in(-\infty,0)\times\operatorname{\mathbb{R}}:\beta<|\alpha|\liminf_{x\rightarrow\infty}\frac{V(x)}{W(x)}\right\\}\subseteq\mathcal{D}_{\varphi}.$

$(ii)$ If $V$ and $W$ satisfy Assumption B, then
$\mathcal{D}_{\varphi}=(-\infty,0)\times\operatorname{\mathbb{R}}$.

###### Proof.

We start with $(i)$, where we assume that
$\bar{x}:=\liminf_{x\rightarrow\infty}\frac{V(x)}{W(x)}\in(0,\infty)$, since
the case where $\bar{x}=\infty$ is treated in $(ii)$. Let
$\alpha\in(-\infty,0)$ be fixed and take some
$\beta\in\operatorname{\mathbb{R}}$ with
$\beta<|\alpha|\liminf_{x\rightarrow\infty}\frac{V(x)}{W(x)}$. We fix
$\delta\in(0,\infty)$ and take $x_{0}\in(0,\infty)$ such that for all
$x\in[x_{0},\infty)$, we have $\frac{V(x)}{W(x)}\geq\bar{x}-\delta$. We then
get

$\displaystyle\int_{x_{0}}^{\infty}e^{\alpha V(x)+\beta W(x)}dx$
$\displaystyle=\int_{x_{0}}^{\infty}\exp\left(\alpha V(x)\left(1+\frac{\beta
W(x)}{\alpha V(x)}\right)\right)dx$
$\displaystyle\leq\int_{x_{0}}^{\infty}\exp\left(\alpha
V(x)\left(1+\frac{\beta}{\alpha(\bar{x}-\delta)}\right)\right)dx$
$\displaystyle\leq\int_{x_{0}}^{\infty}\exp\left(\alpha
V(x)\left(1-\frac{\bar{x}}{\bar{x}-\delta}\right)\right)dx<\infty,$

where the last inequality holds since $1-\frac{\bar{x}}{\bar{x}-\delta}>0$,
for fixed $\delta$. Thus, as claimed $(\alpha,\beta)\in\mathcal{D}_{\varphi}$.
The proof of $(ii)$ works with an analogous argument. ∎

######  Lemma 3.2.

Let $V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be two Orlicz
functions. Then the mappings

$\beta\mapsto\mathbb{E}_{\mu_{\alpha,\beta}}\left[V(X)\right]\quad\text{and}\quad\alpha\mapsto\mathbb{E}_{\mu_{\alpha,\beta}}\left[V(X)\right],\quad(\alpha,\beta)\in\mathcal{D}_{\varphi},$

are well-defined and strictly monotone increasing.

###### Proof.

Let $(\alpha,\beta)\in\mathcal{D}_{\varphi}$, then

$\frac{\partial}{\partial\alpha}\varphi(\alpha,\beta)=\mathbb{E}_{\mu_{\alpha,\beta}}\left[V(X)\right]$

exists (as a derivative of a parameterintegral). We prove in the following
that, for fixed $\alpha\in(-\infty,0)$,
$\beta\mapsto\mathbb{E}_{\mu_{\alpha,\beta}}\left[V(X)\right]$ is strictly
increasing. To that end, let $\beta_{1},\beta_{2}\in\operatorname{\mathbb{R}}$
such that $(\alpha,\beta_{i})\in\mathcal{D}_{\varphi}$ for $i=1,2$ and let
$\beta_{1}<\beta_{2}$. Since $V$ is an Orlicz function (hence $V$ takes only
non-negative values) it follows that
$Z_{\alpha,\beta_{1}}<Z_{\alpha,\beta_{2}}$. Hence, there exists a unique
point $\bar{x}\in(0,\infty)$ such that

$\frac{d\mu_{\alpha,\beta_{1}}}{dx}(\bar{x})=\frac{e^{\alpha
V(\bar{x})+\beta_{1}W(\bar{x})}}{Z_{\alpha,\beta_{1}}}=\frac{e^{\alpha
V(\bar{x})+\beta_{2}W(\bar{x})}}{Z_{\alpha,\beta_{2}}}=\frac{d\mu_{\alpha,\beta_{2}}}{dx}(\bar{x}).$

This implies that

$\displaystyle\frac{e^{\alpha V(x)+\beta_{1}W(x)}}{Z_{\alpha,\beta_{1}}}$
$\displaystyle>\frac{e^{\alpha
V(x)+\beta_{2}W(x)}}{Z_{\alpha,\beta_{2}}}\quad\text{for all }x\in(0,\bar{x})$
$\displaystyle\frac{e^{\alpha V(x)+\beta_{1}W(x)}}{Z_{\alpha,\beta_{1}}}$
$\displaystyle<\frac{e^{\alpha
V(x)+\beta_{2}W(x)}}{Z_{\alpha,\beta_{2}}}\quad\text{for all
}x\in(\bar{x},\infty).$

From that we infer

$\displaystyle\mathbb{E}_{\mu_{\alpha,\beta_{1}}}\left[V(X)\right]$
$\displaystyle=\int_{0}^{\infty}\mathbb{P}_{\alpha,\beta_{1}}\left[V(X)\geq
x\right]dx$
$\displaystyle=2\int_{0}^{\infty}\int_{V^{-1}(x)}^{\infty}\frac{e^{\alpha
V(t)+\beta_{1}W(t)}}{Z_{\alpha,\beta_{1}}}dtdx$
$\displaystyle=2\int_{0}^{V(\bar{x})}\int_{V^{-1}(x)}^{\infty}\frac{e^{\alpha
V(t)+\beta_{1}W(t)}}{Z_{\alpha,\beta_{1}}}dtdx+2\int_{V(\bar{x})}^{\infty}\int_{V^{-1}(x)}^{\infty}\frac{e^{\alpha
V(t)+\beta_{1}W(t)}}{Z_{\alpha,\beta_{1}}}dtdx$
$\displaystyle<2\int_{0}^{V(\bar{x})}\left[\frac{1}{2}-\int_{0}^{V^{-1}(x)}\frac{e^{\alpha
V(t)+\beta_{1}W(t)}}{Z_{\alpha,\beta_{1}}}dt\right]dx+2\int_{V(\bar{x})}^{\infty}\int_{V^{-1}(x)}^{\infty}\frac{e^{\alpha
V(t)+\beta_{2}W(t)}}{Z_{\alpha,\beta_{2}}}dtdx$
$\displaystyle<2\int_{0}^{V(\bar{x})}\left[\frac{1}{2}-\int_{0}^{V^{-1}(x)}\frac{e^{\alpha
V(t)+\beta_{2}W(t)}}{Z_{\alpha,\beta_{2}}}dt\right]dx+2\int_{V(\bar{x})}^{\infty}\int_{V^{-1}(x)}^{\infty}\frac{e^{\alpha
V(t)+\beta_{2}W(t)}}{Z_{\alpha,\beta_{2}}}dtdx$
$\displaystyle=\mathbb{E}_{\mu_{\alpha,\beta_{2}}}\left[V(X)\right].$

An analogous argument shows that
$\alpha\mapsto\mathbb{E}_{\mu_{\alpha,\beta}}\left[V(X)\right]$ is strictly
monotone increasing.

$\bar{x}$$\frac{d\mu_{\alpha,\beta_{1}}}{dx}$$\frac{d\mu_{\alpha,\beta_{2}}}{dx}$
Figure 1: The picture suggests that
$\mathbb{E}_{\mu_{\alpha,\beta_{1}}}[V(X)]<\mathbb{E}_{\mu_{\alpha,\beta_{2}}}[V(X)]$.

∎

######  Lemma 3.3.

Let $R\in(0,\infty)$ and $V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$
be two linearly independent Orlicz functions.

$(i)$ If $V$ and $W$ satisfy Assumption A, then there exists an
$\varepsilon\in(0,\infty)$ such that for all $t\in(m,m+\varepsilon)$, we find
$(\alpha,\beta)\in(-\infty,0)\times(0,\infty)$ such that

$\nabla\varphi(\alpha,\beta)=\left(\begin{array}[]{c}R\\\ t\\\
\end{array}\right).$ (13)

$(ii)$ If $V$ and $W$ satisfy Assumption B, then, for all
$t\in(m,W(V^{-1}(R)))$, we find $(\alpha,\beta)\in(-\infty,0)\times(0,\infty)$
such that (13) holds.

###### Proof.

We start with a few general observation before we prove $(i)$ and $(ii)$. It
is easy to see that
$\varphi:\mathcal{D}_{\varphi}\rightarrow\operatorname{\mathbb{R}}$ is two
times continuous differentiable and, for
$(\alpha,\beta)\in\mathcal{D}_{\varphi}$, the Hessian
$H_{\varphi}(\alpha,\beta)$ of $\varphi$ satisfies

$H_{\varphi}(\alpha,\beta)=\operatorname{Cov}_{\mu_{\alpha,\beta}}\left[\left(V(X),W(X)\right)\right].$

Since $H_{\varphi}(\alpha,\beta)$ is the covariance matrix of the vector
$\left(V(X),W(X)\right)$, where $X\sim\mu_{\alpha,\beta}$ and $V$ and $W$ are
linearly independent on $\operatorname{supp}(X)=\operatorname{\mathbb{R}}$, we
can deduce that $H_{\varphi}(\alpha,\beta)$ is a positive definite matrix. We
define $\alpha^{*}\in(-\infty,0)$ to be the unique point, where (see, e.g.,
Section 1.1. in [3])

$\mathbb{E}_{\mu_{\alpha^{*},0}}[V(X)]=R.$

We are now going to prove $(i)$. By Lemma 3.1, there is a small neighborhood
(in $\operatorname{\mathbb{R}}^{2}$) around $\alpha_{*}$ which is completely
contained in $\mathcal{D}_{\varphi}$. By Lemma 3.2, the mapping
$\beta\mapsto\mathbb{E}_{\mu_{\alpha^{*},\beta}}\left[V(X)\right]$ is strictly
monotone increasing and hence, by continuity, there exists a
$\delta_{1}\in(0,\infty)$ such that for all
$\alpha\in(\alpha^{*}-\delta_{1},\alpha^{*})$, we find a
$\beta^{\prime}\in(0,\infty)$ such that

$\mathbb{E}_{\mu_{\alpha,\beta^{\prime}}}\left[V(X)\right]=R.$

Since $\beta\mapsto\mathbb{E}_{\mu_{\alpha,\beta}}\left[V(X)\right]$ is
strictly mononote increasing, we see that $\beta^{\prime}$ is unique. This
defines a mapping $\alpha\mapsto\beta(\alpha)$ for
$\alpha\in(\alpha^{*}-\delta_{1},\alpha^{*})$ with the property

$\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[V(X)\right]=R.$

The function $\alpha\mapsto\beta(\alpha)$ is continuous and by employing Lemma
3.3, $\beta(\alpha)$ is strictly monotone decreasing on
$(\alpha^{*}-\delta_{1},\alpha^{*})$. By the Lebesgue differentiation theorem,
it follows that
$\beta:(\alpha^{*}-\delta_{1},\alpha^{*})\rightarrow(0,\infty)$ is
differentiable almost everywhere. Since, by construction,
$\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[V(X)\right]\equiv R$, we have
that
$\frac{\partial}{\partial\alpha}\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[V(X)\right]\equiv
0$ and in every point $\alpha\in(\alpha^{*}-\delta_{1},\alpha^{*})$ where
$\beta(\alpha)$ is differentiable, we get by the chain rule

$0=\frac{\partial}{\partial\alpha}\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[V(X)\right]=\mathbb{V}_{\mu_{\alpha,\beta(\alpha)}}\left[V(X)\right]+\beta^{\prime}(\alpha)\operatorname{Cov}_{\mu_{\alpha,\beta(\alpha)}}\left[V(X),W(X)\right]$

We recall that $H_{\varphi}(\alpha,\beta(\alpha))$ is a positive definite
matrix which implies that

$\displaystyle
0<\left(1,\beta^{\prime}(\alpha)\right)H_{\varphi}(\alpha,\beta(\alpha))\left(\begin{array}[]{cc}1\\\
\beta^{\prime}(\alpha)\end{array}\right)$
$\displaystyle=\beta^{\prime}(\alpha)\left(\operatorname{Cov}_{\mu_{\alpha,\beta(\alpha)}}\left[V(X),W(X)\right]+\beta^{\prime}(\alpha)\mathbb{V}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]\right)$
$\displaystyle=\beta^{\prime}(\alpha)\left(\frac{\partial}{\partial\alpha}\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]\right).$

Since $\beta^{\prime}(\alpha)\in(-\infty,0)$, we have
$\frac{\partial}{\partial\alpha}\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]\in(-\infty,0)$.
This means, in almost every point
$\alpha\in(\alpha^{*}-\delta_{1},\alpha^{*})$,
$\frac{\partial}{\partial\alpha}\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]$
exists and is negative. Thus, the mapping

$\alpha\mapsto\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]$

is continous and strictly monotone decreasing on
$(\alpha^{*}-\delta_{1},\alpha^{*})$. Since
$m=\mathbb{E}_{\mu_{\alpha^{*},0}}\left[W(X)\right]$, it follows that

$\varepsilon:=\sup_{\alpha\in(\alpha^{*}-\delta_{1},\alpha^{*})}\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]-m\in(0,\infty),$

and for every $t\in(m,m+\varepsilon)$, we find a pair
$(\alpha,\beta)\in(-\infty,0)\times(0,\infty)$ such that

$\nabla\varphi(\alpha,\beta)=\left(\begin{array}[]{c}\mathbb{E}_{\mu_{\alpha,\beta}}\left[V(X)\right]\\\
\mathbb{E}_{\mu_{\alpha,\beta}}\left[W(X)\right]\end{array}\right)=\left(\begin{array}[]{c}R\\\
t\end{array}\right),$

which proves $(i)$.

$-\infty$$\alpha_{*}$$\mathcal{D}_{\varphi}$$\beta(\alpha)$ Figure 2: In
$(i)$, we are able to construct $\beta(\alpha)$ locally to the left of
$\alpha_{*}$.

Now we are going to prove $(ii)$. By Lemma 3.1, it follows that
$\mathcal{D}_{\varphi}=(-\infty,0)\times\operatorname{\mathbb{R}}$, hence for
every $\alpha\in(-\infty,\alpha^{*})$, we find an unique
$\beta=\beta(\alpha)\in(0,\infty)$ such that

$\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[V(X)\right]=R.$

The mapping $\alpha\mapsto\beta(\alpha)$ is strictly monotone increasing
(hence differentiable almost everywhere) and analogously to the first part of
this prove, one has that

$\alpha\mapsto\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]$

is continuous and strictly monotone decreasing. We define

$x:=\sup_{\alpha\in(-\infty,\alpha^{*})}\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]=\lim_{\alpha\rightarrow-\infty}\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right].$

By the continuity of
$\alpha\mapsto\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]$, for
all $t\in(m,x)$ we find $(\alpha,\beta)\in(-\infty,0)\times(0,\infty)$ with

$\nabla\varphi(\alpha,\beta)=\left(\begin{array}[]{c}\mathbb{E}_{\mu_{\alpha,\beta}}\left[V(X)\right]\\\
\mathbb{E}_{\mu_{\alpha,\beta}}\left[W(X)\right]\end{array}\right)=\left(\begin{array}[]{c}R\\\
t\end{array}\right).$

In the rest of this proof, we will show that $x=W(V^{-1}(R))$. Let
$\mathcal{M}=(\mu_{\alpha,\beta(\alpha)})_{\alpha\in(-\infty,\alpha^{*})}$,
then $\mathcal{M}$ is a tight family of probability measures (otherwise
$R\not\equiv\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[V(X)\right]$), thus
every sub-family of $\mathcal{M}$ is also tight. For any sequence
$(\alpha_{n})_{n\in\operatorname{\mathbb{N}}}$ with
$\lim_{n\rightarrow\infty}\alpha_{n}=-\infty$, we have that

$x=\lim_{n\rightarrow\infty}\mathbb{E}_{\mu_{\alpha_{n},\beta(\alpha_{n})}}\left[W(X)\right].$

Assume there exists a sequence $(\alpha_{n})_{n\in\operatorname{\mathbb{N}}}$
tending to $-\infty$ with

$\lim_{n\rightarrow\infty}\mathbb{V}_{\mu_{\alpha_{n},\beta(\alpha_{n})}}\left[V(X)\right]=0.$

Since $(\mu_{\alpha_{n},\beta(\alpha_{n})})_{n\in\operatorname{\mathbb{N}}}$
is tight, we find a weakly convergent subsequence
$(\mu_{\alpha_{n_{k}},\beta(\alpha_{n_{k}})})_{k\in\operatorname{\mathbb{N}}}$
with (weak) limit $\mu_{\infty}$. We trivially have that

$\mathbb{E}_{\mu_{\infty}}\left[V(X)\right]=R.$

But, since $\mathbb{V}_{\mu_{\infty}}\left[V(X)\right]=0$ and by the symmetry
of $\mu_{\infty}$, it follows that

$\mu_{\infty}=\frac{1}{2}\delta_{-V^{-1}(R)}+\frac{1}{2}\delta_{V^{-1}(R)}.$

This gives us that $x=W(V^{-1}(R))$. Now consider the case where
$\inf_{\alpha\in(-\infty,\alpha^{*})}\mathbb{V}_{\mu_{\alpha,\beta(\alpha)}}\left[V(X)\right]\in(0,\infty)$.
We know that $x\in(0,\infty)$, otherwise

$\lim_{\alpha\rightarrow-\infty}\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]=\infty,$

which is a contradiction to
$R=\lim_{\alpha\rightarrow-\infty}\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[V(X)\right]$.
Let $A\subseteq(-\infty,\alpha^{*})$ be the set where $\beta^{\prime}$

(and hence also
$\frac{\partial}{\partial\alpha}\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]$)
exists. We claim that there exists a subsequence
$(\alpha_{n})_{n\in\operatorname{\mathbb{N}}}\subseteq A$ such that

$\lim_{n\rightarrow\infty}\frac{\partial}{\partial\alpha}\mathbb{E}_{\mu_{\alpha_{n},\beta(\alpha_{n})}}\left[W(X)\right]=0.$

If such a sequence $(\alpha_{n})_{n\in\operatorname{\mathbb{N}}}\subseteq A$
would not exist, then there is an $\varepsilon\in(0,\infty)$ and
$\alpha^{\prime}=\alpha^{\prime}(\varepsilon)\in(-\infty,\alpha^{*})$ such
that for all $\alpha\in(-\infty,\alpha^{\prime})$ we have

$\frac{\partial}{\partial\alpha}\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]\leq-\varepsilon.$

By integrating over the infinite interval $(-\infty,\alpha^{\prime})$ we would
now get $x=\infty$, which is a contradiction. Thus, we can assume that there
exists a sequence $(\alpha_{n})_{n\in\operatorname{\mathbb{N}}}\subseteq A$
with

$\lim_{n\rightarrow\infty}\frac{\partial}{\partial\alpha}\mathbb{E}_{\mu_{\alpha_{n},\beta(\alpha_{n})}}\left[W(X)\right]=0.$
(14)

We claim that, along this sequence
$(\alpha_{n})_{n\in\operatorname{\mathbb{N}}}$, we have
$\sup_{n\in\operatorname{\mathbb{N}}}\frac{|\beta(\alpha_{n})|}{\alpha_{n}}\in(0,\infty)$.
If this was not the case, then, along a suitable subsequence
$(\alpha_{n_{k}})_{k\in\operatorname{\mathbb{N}}}\subseteq(\alpha_{n})_{n\in\operatorname{\mathbb{N}}}$,
we would have that
$\lim_{k\rightarrow\infty}\frac{|\beta(\alpha_{n_{k}})|}{\alpha_{n_{k}}}=\infty$.
Let $M\in(0,\infty)$ be some large number and denote $x_{k}\in(0,\infty)$ to
be the maximizer of $x\mapsto\alpha_{n_{k}}V(x)+\beta(\alpha_{n_{k}})W(x)$ on
$(0,M)$ and let $y_{k}\in(0,\infty)$ be the minimizer of
$y\mapsto\alpha_{n_{k}}V(y)+\beta(\alpha_{n_{k}})W(y)$ on $(2M,3M)$. Then,
since $V$ and $W$ are Orlicz functions, there exist constants
$c,C\in(0,\infty)$ such that $V(y_{k})-V(x_{k})\leq C$ and
$W(y_{k})-W(x_{k})\geq c$ for all $k\in\operatorname{\mathbb{N}}$. This leads
to

$\displaystyle\mu_{\alpha_{n_{k}},\beta(\alpha_{n_{k}})}\left[(0,M)\right]$
$\displaystyle=\frac{2\int_{0}^{M}e^{\alpha_{n_{k}}V(x)+\beta(\alpha_{n_{k}})W(x)}dx}{Z_{\alpha_{n_{k}}}}$
$\displaystyle\leq\frac{2Me^{\alpha_{n_{k}}V(x_{k})+\beta(\alpha_{n_{k}})W(x_{k})}}{Z_{\alpha_{n_{k}}}}$
$\displaystyle\leq
e^{-\alpha_{n_{k}}\left(V(y_{k})-V(x_{k})\right)-\beta(\alpha_{n_{k}})\left(W(y_{k})-W(x_{k})\right)}$
$\displaystyle\leq
e^{-\alpha_{n_{k}}C-\beta(\alpha_{n_{k}})c}\stackrel{{\scriptstyle
k\rightarrow\infty}}{{\longrightarrow}}0.$

This means, for every fixed $M$, the
$\mu_{\alpha_{n_{k}},\beta(\alpha_{n_{k}})}$-probability of $(0,M)$ tends to
$0$ as $k\rightarrow\infty$. This contradicts the construction of
$\beta(\alpha_{n_{k}})$, where we have

$R=\mathbb{E}_{\mu_{\alpha_{n_{k}},\beta(\alpha_{n_{k}})}}\left[V(X)\right].$

Thus,
$\left(\frac{\beta(\alpha_{n})}{\alpha_{n}}\right)_{n\in\operatorname{\mathbb{N}}}$
is a bounded sequence and we find a convergent subsequence
$(\gamma_{n}^{(1)})_{n\in\operatorname{\mathbb{N}}}\subseteq\left(\alpha_{n}\right)_{n\in\operatorname{\mathbb{N}}}$
with limit $-\lambda\in(-\infty,0)$. Further, by de L’Hospital’s rule, we have
that $-\lambda=\lim_{n\rightarrow\infty}\beta^{\prime}(\alpha_{n})$.

Since $\mathcal{M}$ is tight, we find another subsequence
$(\gamma_{n}^{(2)})_{n\in\operatorname{\mathbb{N}}}\subseteq(\gamma_{n}^{(1)})_{n\in\operatorname{\mathbb{N}}}$
and a distribution $\mu_{\infty}$ such that
$\mu_{\gamma_{n}^{(2)},\beta(\gamma_{n}^{(2)})}\longrightarrow\mu_{\infty}$ as
$n\rightarrow\infty$ (we understand this limit as weak limit of
distributions). We claim that for all $x\in\operatorname{\mathbb{R}}$, we have
$V(x)\geq\lambda W(x)$. Assume this was not the case, i.e., we find a small
interval $I=(y-\delta,y+\delta)\subseteq\operatorname{\mathbb{R}}$ and some
$\varepsilon\in(0,\infty)$ such that for all $x\in(y-\delta,y+\delta)$

$V(x)-\lambda W(x)\leq-\varepsilon.$

Then (since
$\frac{\beta(\gamma_{n}^{(2)})}{\gamma_{n}^{(2)}}\leq-\lambda+\frac{\varepsilon}{2}\left(\sup_{x\in
I}W(x)\right)^{-1}$, if $n$ is sufficiently large)

$\displaystyle Z_{\gamma_{n}^{(2)},\beta(\gamma_{n}^{(2)})}$
$\displaystyle=\int_{\operatorname{\mathbb{R}}}e^{\gamma_{n}^{(2)}V(x)+\beta(\gamma_{n}^{(2)})W(x)}dx$
$\displaystyle=\int_{\operatorname{\mathbb{R}}}\exp\left(\gamma_{n}^{(2)}\left(V(x)+\frac{\beta(\gamma_{n}^{(2)})}{\gamma_{n}^{(2)}}W(x)\right)\right)dx$
$\displaystyle\geq\int_{I}\exp\left(\gamma_{n}^{(2)}\left(V(x)+\frac{\beta(\gamma_{n}^{(2)})}{\gamma_{n}^{(2)}}W(x)\right)\right)dx$
$\displaystyle\geq\int_{I}\exp\left(-\frac{\varepsilon\gamma_{n}^{(2)}}{2}\right)dx\longrightarrow\infty,\quad
n\rightarrow\infty.$

From that we deduce

$\mu_{\infty}\left[x\in\operatorname{\mathbb{R}}:V(x)-\lambda W(x)\leq
0\right]=1\quad\text{and}\quad\mu_{\infty}\left[x\in\operatorname{\mathbb{R}}:V(x)-\lambda
W(x)\leq-\varepsilon\right]>0.$

This means, for a random variable $X\sim\mu_{\infty}$, we have
$V(X)\leq\lambda W(X)$ almost surely and with positive probability we have
$V(X)<\lambda W(X)$. By the very construction of
$(\alpha_{n})_{n\in\operatorname{\mathbb{N}}}$, of which
$(\gamma_{n}^{(2)})_{n\in\operatorname{\mathbb{N}}}$ is a subsequence, we have
by (14)

$0=\operatorname{Cov}_{\mu_{\infty}}\left[V(X),W(X)\right]-\lambda\mathbb{V}_{\mu_{\infty}}\left[W(X)\right].$

Recall that this quantity is a (non-trivial) quadratic form of the covariance
matrix $C\in\operatorname{\mathbb{R}}^{2\times 2}$ of
$\left(V(X),W(X)\right)$. This means $V(X)$ and $W(X)$ are linearly dependent,
or in other words, there exists a $\nu\in(0,\infty)$ such that $V(X)=\nu W(X)$
almost surely (although we assumed that $V$ and $W$ are linearly independent
on $\operatorname{\mathbb{R}}$, this does not necessarily mean that $V$ and
$W$ are linearly independent on $\operatorname{supp}(X)$). Since
$V(X)\leq\lambda W(X)$ almost surely, we thus have $\nu\leq\lambda$. This
leads to

$\displaystyle 0$
$\displaystyle=\operatorname{Cov}_{\mu_{\infty}}\left[V(X),W(X)\right]-\lambda\mathbb{V}_{\mu_{\infty}}\left[W(X)\right]$
$\displaystyle=\frac{1}{\nu}\mathbb{V}_{\mu_{\infty}}\left[V(X)\right]-\frac{\lambda}{\nu^{2}}\mathbb{V}_{\mu_{\infty}}\left[V(X)\right].$

Since by assumption $\mathbb{V}_{\mu_{\infty}}\left[V(X)\right]\in(0,\infty)$,
we get $\nu=\lambda$. But this is a contradiction, since, on the one hand, we
have $V(X)=\lambda W(X)$ almost surely, and on the other hand, $V(X)<\lambda
W(X)$ with positive probability. This means, $\lambda$ has the property that
$V(x)\geq\lambda W(x)$ for all $x\in\operatorname{\mathbb{R}}$. In particular,
it is true that

$\lambda\leq\frac{R}{W(V^{-1}(R))}.$

We now get (using that $\mathbb{E}_{\mu_{\infty}}\left[V(X)\right]=R$ and
$W(X)=\frac{1}{\lambda}V(X)$)

$\displaystyle x$
$\displaystyle=\mathbb{E}_{\mu_{\infty}}\left[W(X)\right]=\frac{1}{\lambda}\mathbb{E}_{\mu_{\infty}}\left[V(X)\right]\geq
W(V^{-1}(R)).$

This finishes the proof.

$\mathcal{D}_{\varphi}$$\alpha^{*}$$\beta(\alpha)$$-\infty$ Figure 3: In
$(ii)$, we construct $\beta(\alpha)$ for all $\alpha\in(-\infty,\alpha^{*})$.
The gray-shaded area indicates the region $\mathcal{D}_{\varphi}$.

∎

######  Remark 3.4.

When looking at the proof of Lemma 3.3, one is tempted to think that assuming

$\liminf_{x\rightarrow\infty}\frac{V(x)}{W(x)}>0$

should be sufficient to push $\varepsilon$ as far as possible, i.e.,
$\varepsilon=W(V^{-1}(R))-m$. But this is not the case! If we set
$V(x):=x^{4}+x^{2}$ and $W(x):=x^{4}$, then $V$ and $W$ are linearly
independent and we have

$\liminf_{x\rightarrow\infty}\frac{V(x)}{W(x)}=1>0.$

It is clear that for $R=10^{3}$ and for $\alpha=-1$, we have

$\sup_{\beta:(\alpha,\beta)\in\mathcal{D}_{\varphi}}\mathbb{E}_{\mu_{-1,\beta}}\left[V(X)\right]=\mathbb{E}_{\mu_{-1,1}}\left[V(X)\right]<R.$

Thus, the construction of $\beta(\alpha)$ as in the proof of Lemma 3.10 $(ii)$
is possible only locally. We further have

$\lim_{\alpha\rightarrow-\infty}\sup_{\beta:(\alpha,\beta)\in\mathcal{D}_{\varphi}}\mathbb{E}_{\mu_{\alpha,\beta}}\left[W(X)\right]=\lim_{\alpha\rightarrow-\infty}\mathbb{E}_{\mu_{\alpha,|\alpha|}}\left[W(X)\right]=0,$

destroying the hope of maximizing $\varepsilon$ by letting $\alpha$ tend to
$-\infty$. A better understanding of the image of $\nabla^{-1}(\varphi)$ would
help to characterize the precise value of $\varepsilon$ in both, Theorem A and
Theorem C.

Heuristically speaking, the goal of Lemma 3.3 was to maximize
$\mathbb{E}_{\mu_{\alpha,\beta}}\left[W(X)\right]$ under the constraint
$\mathbb{E}_{\mu_{\alpha,\beta}}\left[V(X)\right]\equiv R$. It turned out that
this can be achieved by letting $\alpha$ tend to $-\infty$ and choosing
$\beta=\beta(\alpha)$ such that
$\mathbb{E}_{\mu_{\alpha,\beta}}\left[V(X)\right]\equiv R$. In the proof of
Theorem C, we need to choose a pair
$(\alpha,\beta)\in\operatorname{\mathbb{R}}^{2}$ such that

$\mathbb{E}_{\mu_{\alpha,\beta}}\left[V(X)\right]\equiv
R\quad\text{and}\quad\mathbb{E}_{\mu_{\alpha,\beta}}\left[W(X)\right]\text{ is
"small"}.$

This is achieved through the following Lemma.

######  Lemma 3.5.

Let $V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be two linearly
independent Orlicz functions that satisfy Assumption A and let
$R\in(0,\infty)$. Then, there exists an $\varepsilon\in(0,\infty)$ such that
for all $t\in(m-\varepsilon,m)$, we find a pair
$(\alpha,\beta)\in(-\infty,0)^{2}$ such that

$\nabla\varphi(\alpha,\beta)=\left(\begin{array}[]{c}R\\\
t\end{array}\right).$ (15)

###### Proof.

Let $\alpha_{*}\in(-\infty,0)$ be the unique solution to

$\mathbb{E}_{\mu_{\alpha_{*},0}}\left[V(X)\right]=R.$

On the other hand, analogously, we can find a unique $\beta_{*}\in(-\infty,0)$
such that

$\mathbb{E}_{\mu_{0,\beta_{*}}}\left[V(X)\right]=R.$

Thus, for any $\alpha\in(\alpha_{*},0)$, we can find a unique
$\beta=\beta(\alpha)\in(-\infty,0)$ such that

$\alpha\mapsto\mathbb{E}_{\mu_{0,\beta_{*}}}\left[V(X)\right]\equiv R.$

$\alpha^{*}$$\beta^{*}$$\beta(\alpha)$$\mathcal{D}_{\varphi}$$-\infty$ Figure
4: In Lemma 3.5, the construction of $\beta(\alpha)$ changes slightly compared
to Lemma 3.3.

Analogously to the proof of Lemma 3.3, we have that $\beta(\alpha)$ is
strictly decreasing and hence almost surely differentiable. Moreover, it holds
that

$\alpha\mapsto\mathbb{E}_{\mu_{\alpha,\beta(\alpha)}}\left[W(X)\right]$

is strictly decreasing on $(\alpha_{*},0)$ and thus, by defining

$\varepsilon:=m-\mathbb{E}_{\mu_{0,\beta_{*}}}\left[W(X)\right]\in(0,\infty),$

the claim follows. ∎

###  3.2 Establishing an Edgeworth expansion

In this subsection we prove that, for linearly independent $V,W$, the sequence
of random variables

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left(V(X_{i}),W(X_{i})\right),$ (16)

where $(X_{i})_{i\in\operatorname{\mathbb{N}}}$ is an iid sequence with
$X_{1}\sim\mu_{\alpha,\beta}$, satisfies an Edgeworth expansion. To that end,
we need to establish a Cramér-type bound for the characteristic function
$\varphi:\operatorname{\mathbb{R}}^{2}\rightarrow\mathbb{C}$ of (16) given by

$\varphi(t,s):=\mathbb{E}_{\mu_{\alpha,\beta}}\left[\exp\left(itV(X)+isW(X)\right)\right],\quad(t,s)\in\operatorname{\mathbb{R}}^{2}.$

In the proof of Lemma 3.10 we show that

$\limsup_{||(t,s)||_{2}\rightarrow\infty}\left|\varphi(t,s)\right|=\limsup_{||(t,s)||_{2}\rightarrow\infty}\left|\frac{1}{Z_{\alpha,\beta}}\int_{\operatorname{\mathbb{R}}}e^{\alpha
V(x)+\beta W(x)}e^{itV(x)+isW(x)}dx\right|<1.$ (17)

In the parlance of probabilistic number theory, (17) is satisfied if the
oscillatory integral given by the characteristic function $\varphi$ shows at
least a very small cancellation. Showing (17) is not straight forward, since
our sequence in (16) is a sequence of random vectors in
$\operatorname{\mathbb{R}}^{2}$. In the classical theory (see, e.g., [9,
Section 2]), we would like to see that the distribution of

$\left(V(X_{1}),W(X_{1})\right),\quad X_{1}\sim\mu_{\alpha,\beta},$

has a non-degenerate (w.r.t. the $2$-dimensional Lebesgue measure) component.
Since this particular sequence is concentrated on the $1$-dimensional manifold

$\mathcal{M}:=\left\\{\left(V(x),W(x)\right)|x\in\operatorname{\mathbb{R}}\right\\},$

the distribution of the random vector in (16) is degenerated in the
$2$-dimensional sense. So we need to tackle the limit in (17) directly. The
classical tool to deal with such integrals is Van der Corput’s lemma (see
Proposition 3.6). In our case, we are dealing with a phase function of the
form $tV+sW$, where $(t,s)\in\operatorname{\mathbb{R}}^{2}$ and $V,W$ are
either Orlicz functions or $V(x)=|x|^{p}$ and $W(x)=|x|^{q}$ with
$p,q\in(0,1)$ and $p\neq q$. In many cases, the function $x\mapsto
tV(x)+sW(x)$ lacks the necessary smoothness and may also have many inflection
points over $\operatorname{\mathbb{R}}$. Thus, we are not able to directly
apply Van der Corput’s lemma. Our idea is to find a family of intervals
$J=J(t,s)\subseteq\operatorname{\mathbb{R}}$ such that the function $tV+sW$
restricted on $J$ is $\mathcal{C}^{2}$ and is strictly monotone and has no
inflection points on $J$. On such an interval we can apply Van der Corput’s
lemma to see at least some amount of cancellation in (17). In order for
everything to work out nicely, we need to impose more regularity on $V$ and
$W$, which is the reason for our notion of an Orlicz function.

######  Proposition 3.6.

(Van der Corput’s lemma) Let $J=(a,b)\subseteq\operatorname{\mathbb{R}}$ be an
interval and $f:J\rightarrow\operatorname{\mathbb{R}}$ be a convex or concave
function which is continuously differentiable on $J$ and assume that
$|f^{\prime}(x)|>0$ for all $x\in J$. Moreover, let
$\phi:J\rightarrow\operatorname{\mathbb{R}}$ be a $\mathcal{C}^{1}(J)$
function. Then, there exists an universal constant $c\in(0,\infty)$ such that
for all $\lambda\in\operatorname{\mathbb{R}}\backslash\\{0\\}$ , it holds

$\left|\int_{a}^{b}\phi(x)e^{i\lambda f(x)}dx\right|\leq c\left(\max_{x\in
J}|\phi(x)|+\int_{J}|\phi^{\prime}(x)|dx\right)\frac{1}{|\lambda|}.$

######  Proposition 3.7.

Let $V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be two Orlicz
functions or let $V(x)=|x|^{p}$ and $W(x)=|x|^{q}$ with $p,q\in(0,1)$ and
$p\neq q$. For $\lambda\in[-1,1]$, let $\mathcal{M}_{\lambda}$ denote the set
of points in $\operatorname{\mathbb{R}}$ where the function

$x\mapsto V(x)+\lambda W(x)$

changes its curvature. Then $\overline{\mathcal{M}_{\lambda}}$ cannot contain
an interval.

###### Proof.

The proof is done by contradiction. We assume there exists a
$\bar{\lambda}\in\operatorname{\mathbb{R}}$ and a non-trivial interval
$I\subseteq\operatorname{\mathbb{R}}$ such that the set of inflection points
of $x\mapsto V(x)+\bar{\lambda}W(x)$ is dense in $I$. Then, by our definition
of an Orlicz function, we find a non-trivial subinterval $J\subseteq I$, where
both $V$ and $W$ are $\mathcal{C}^{2}$. On $J$, the set of inflection points
of $V+\bar{\lambda}W$ is still dense and thus
$V^{\prime\prime}+\bar{\lambda}W^{\prime\prime}$ is not continuous on $J$,
which is an immediate contradiction. ∎

######  Proposition 3.8.

Let $V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be two Orlicz
functions or let $V(x)=|x|^{p}$ and $W(x)=|x|^{q}$ with $p,q\in(0,1)$ and
$p\neq q$. Then, there exists a compact interval
$I\subseteq\operatorname{\mathbb{R}}$, some $\delta\in(0,\infty)$ and a family
of intervals $(J_{\lambda})_{\lambda\in[-1,1]}$ with $J_{\lambda}\subseteq I$
for all $\lambda\in[-1,1]$ and
$\inf_{\lambda\in[-1,1]}|J_{\lambda}|\geq\delta$ such that, for
$\lambda\in[-1,1]$, the function

$x\mapsto V(x)+\lambda W(x)$

is $\mathcal{C}^{2}$, strictly monotone and does not change its curvature on
$J_{\lambda}$.

###### Proof.

First we show that there exists a compact interval
$I\subseteq\operatorname{\mathbb{R}}$ and some $\delta^{\prime}\in(0,\infty)$
such that, for all $\lambda\in[-1,1]$, there exists an interval
$J_{\lambda}^{\prime}\subseteq I$ with $x\mapsto V(x)+\lambda W(x)$ being
strictly monotone throughout $J_{\lambda}^{\prime}$ and
$|J_{\lambda}^{\prime}|\geq\delta^{\prime}$. We claim that there exists an
interval $I\subseteq\operatorname{\mathbb{R}}$ such that for all
$\lambda\in[-1,1]$, the function $V+\lambda W$ is not constant on $I$. If such
an interval $I$ would not exist, for every $M\in(0,\infty)$, we would find a
$\lambda\in[-1,1]$ such that $V+\lambda W$ is constant (and hence $0$) on
$[-M,M]$, which is a contradiction to the linear independence of $V$ and $W$.
Since, for every $\lambda\in[-1,1]$, the function $V+\lambda W$ is not
constant on $I$, we can define $J_{\lambda}^{\prime}\subseteq I$ to be the
largest subinterval of $I$ such that $V+\lambda W$ is either strictly monotone
increasing on $J_{\lambda}^{\prime}$ or strictly monotone decreasing on
$J_{\lambda}^{\prime}$. Moreover, we claim that there exists a
$\delta^{\prime}\in(0,\infty)$ such that
$\inf_{\lambda\in[-1,1]}|J_{\lambda}^{\prime}|\geq\delta^{\prime}$. Assume
that such a positive $\delta^{\prime}$ with
$\inf_{\lambda\in[-1,1]}|J_{\lambda}^{\prime}|\geq\delta^{\prime}$ does not
exist. Then, for every $n\in\operatorname{\mathbb{N}}$, we find a
$\lambda_{n}\in[-1,1]$ such that $|J_{\lambda_{n}}^{\prime}|<\frac{1}{n}$.
Since $(\lambda_{n})_{n\in\operatorname{\mathbb{N}}}$ is a bounded sequence,
there exists a convergent subsequence
$(\lambda_{n_{k}})_{k\in\operatorname{\mathbb{N}}}$ with limit
$\bar{\lambda}\in[-1,1]$. By construction, $V+\bar{\lambda}W$ is constant on
$I$ which is an immediate contradiction. Thus, there exists a
$\delta^{\prime}\in(0,\infty)$ with
$\inf_{\lambda\in[-1,1]}|J_{\lambda}^{\prime}|\geq\delta^{\prime}$.

Now, for all $\lambda\in[-1,1]$, let $J_{\lambda}\subseteq
J_{\lambda}^{\prime}$ be the largest subinterval of $J_{\lambda}$ such that
$x\mapsto V(x)+\lambda W(x)$ does not change its curvature throughout
$J_{\lambda}$. By Proposition 3.7 it follows that there must exist a
$\delta\in(0,\delta^{\prime})$ such that
$\inf_{\lambda\in[-1,1]}|J_{\lambda}|\geq\delta$.

The case where $V(x)=|x|^{p}$ and $W(x)=|x|^{q}$ can be argued similarly. ∎

######  Proposition 3.9.

Let
$\left((t_{n},s_{n})\right)_{n\in\operatorname{\mathbb{N}}}\subseteq\operatorname{\mathbb{R}}^{2}$
be a sequence with $|t_{n}|\geq|s_{n}|$ for all
$n\in\operatorname{\mathbb{N}}$ and $\lim_{n\rightarrow\infty}|t_{n}|=\infty$.
Further, let $V,W:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be two
linearly independent Orlicz functions or $V(x)=|x|^{p}$ and $W(x)=|x|^{q}$
with $p,q\in(0,1)$ and $p\neq q$. Then, there exists an $\varepsilon\in(0,1)$
and some sequence $(\delta_{n})_{n\in\operatorname{\mathbb{N}}}$ with
$\delta_{n}\rightarrow 0$ such that

$\left|\frac{1}{Z_{\alpha,\beta}}\int_{\operatorname{\mathbb{R}}}e^{\alpha
V(x)+\beta W(x)}e^{it_{n}V(x)+is_{n}W(x)}\right|\leq
1-\varepsilon+\delta_{n},\quad n\in\operatorname{\mathbb{N}}.$

###### Proof.

We prove this proposition by applying Van der Corput’s lemma (see Proposition
3.6). For $n\in\operatorname{\mathbb{N}}$, define
$\lambda_{n}:=\frac{s_{n}}{t_{n}}$ (we can assume that $t_{n}\neq 0$ for all
$n\in\operatorname{\mathbb{N}}$, since
$\lim_{n\rightarrow\infty}|t_{n}|=\infty$). Then $\lambda_{n}\in[-1,1]$ for
all $n\in\operatorname{\mathbb{N}}$ by assumption. By Proposition 3.8, there
exists a $\delta\in(0,\infty)$, a compact interval
$I\subseteq\operatorname{\mathbb{R}}$ and a sequence of intervals
$(J_{\lambda_{n}})_{n\in\operatorname{\mathbb{N}}}$ with
$\inf_{n\in\operatorname{\mathbb{N}}}|J_{\lambda_{n}}|\geq\delta$ and
$J_{\lambda_{n}}\subseteq I$ for all $n\in\operatorname{\mathbb{N}}$ such that
the function

$x\mapsto V(x)+\lambda_{n}W(x)$

is $\mathcal{C}^{2}$, strictly monotone and does not change its curvature on
$J_{\lambda_{n}}$. By Proposition 3.6, we thus get

$\left|\frac{1}{Z_{\alpha,\beta}}\int_{J_{\lambda_{n}}}e^{\alpha V(x)+\beta
W(x)}e^{it_{n}\left(V(x)+\lambda_{n}W(x)\right)}dx\right|\leq\frac{c_{n}}{|t_{n}|},$

where we have

$\displaystyle c_{n}$ $\displaystyle=c\left(\max_{x\in
J_{\lambda_{n}}}\left[e^{\alpha V(x)+\beta
W(x)}\right]+\int_{J_{\lambda_{n}}}\left|\alpha V^{\prime}(x)+\beta
W^{\prime}(x)\right|e^{\alpha V(x)+\beta W(x)}dx\right).$

The construction of $J_{\lambda_{n}}$ implies that all $c_{n}$ are uniformly
bounded by some absolute constant $C\in(0,\infty)$. We thus have

$\left|\frac{1}{Z_{\alpha,\beta}}\int_{J_{\lambda_{n}}}e^{\alpha V(x)+\beta
W(x)}e^{it_{n}\left(V(x)+\lambda_{n}W(x)\right)}dx\right|\leq\frac{C}{|t_{n}|}=:\delta_{n}.$
(18)

Since all intervals $J_{\lambda_{n}}$ are contained in a fixed compact
interval $I\subseteq\operatorname{\mathbb{R}}$ and all $J_{\lambda_{n}}$ have
length of at least $\delta\in(0,\infty)$, there must exist an
$\varepsilon\in(0,1)$ such that

$\left|\frac{1}{Z_{\alpha,\beta}}\int_{J_{\lambda_{n}}^{c}}e^{\alpha
V(x)+\beta W(x)}e^{it_{n}\left(V(x)+\lambda_{n}W(x)\right)}dx\right|\leq
1-\varepsilon.$ (19)

The estimates in (18) and (19) now show the claim. ∎

######  Lemma 3.10.

Let $V,W:\operatorname{\mathbb{R}}\rightarrow\operatorname{\mathbb{R}}$ be two
linearly independent Orlicz functions or $V(x)=|x|^{p}$ and $W(x)=|x|^{q}$
with $p,q\in(0,1)$ and $p\neq q$. Let
$(\alpha,\beta)\in\operatorname{\mathbb{R}}^{2}$ such that
$\mu_{\alpha,\beta}$ exists and let
$\left(X_{i}\right)_{i\in\operatorname{\mathbb{N}}}$ be an iid sequence with
$X_{1}\sim\mu_{\alpha,\beta}$. We assume that there exists a
$k\in\operatorname{\mathbb{N}}_{\geq 2}$ with

$\mathbb{E}_{\mu_{\alpha,\beta}}\left[\left|\left|\left(V(X),W(X)\right)\right|\right|_{2}^{k}\right]<\infty.$
(20)

We set
$(R,t):=\left(\mathbb{E}\left[V(X_{1})\right],\mathbb{E}\left[W(X_{1})\right]\right)$
and, for $n\in\operatorname{\mathbb{N}}$, we denote the distribution function
of the vector

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left(V(X_{i})-R,W(X_{i})-t\right)$

as $F_{n}$. Then, for any $(x,y)\in\operatorname{\mathbb{R}}^{2}$, we can
write

$F_{n}(x,y)=G^{\Sigma}(x,y)+\sum_{j=1}^{k}n^{-j/2}F_{j}(x,y)+R_{n}(x,y),$

where the remainder $R_{n}$ satisfies $R_{n}(x,y)=o(n^{-k/2})$, uniformly in
$(x,y)\in\operatorname{\mathbb{R}}^{2}$. $G^{\Sigma}$ is the distribution
function of a normal distribution with mean
$0\in\operatorname{\mathbb{R}}^{2}$ and covariance matrix
$\Sigma\in\operatorname{\mathbb{R}}^{2\times 2}$ given by

$\Sigma=\operatorname{Cov}\left[\left(V(X_{1}),W(X_{1})\right)\right].$

Moreover, the functions $F_{j}$, for $j=1,\ldots,k$, can be computed via

$F_{j}(x,y)=\int_{-\infty}^{x}\int_{-\infty}^{y}g^{\Sigma}(t,s)\pi_{j}(t,s)dtds,\quad(x,y)\in\operatorname{\mathbb{R}}^{2},$

where $g^{\Sigma}$ is the density of $G^{\Sigma}$ and $\pi_{j}$ is a
polynomial in two variables of degree at most $3j$.

###### Proof.

Since $V,W$ are linearly independent, the covariance matrix
$\Sigma:=\operatorname{Cov}\left(V(X_{1}),W(X_{1})\right)\in\operatorname{\mathbb{R}}^{2\times
2}$ (which exists by (20)) is positive definite. Thus, the result is a
consequence of the elaborations in [9, Chapter 2], provided the assumption

$\limsup_{||(t,s)||_{2}\rightarrow\infty}\left|\mathbb{E}_{\mu_{\alpha,\beta}}\left[e^{itV(X)+isW(X)}\right]\right|<1$

is satisfied. To that end, let
$\left((t_{n},s_{n})\right)_{n\in\operatorname{\mathbb{N}}}$ be a sequence of
real numbers with $||(t_{n},s_{n})||_{2}\rightarrow\infty$ as
$n\rightarrow\infty$, where we need to show that

$\limsup_{n\rightarrow\infty}\left|\mathbb{E}_{\mu_{\alpha,\beta}}\left[e^{it_{n}V(X)+is_{n}W(X)}\right]\right|<1.$
(21)

We define two sequences of natural numbers
$(n_{k})_{k\in\operatorname{\mathbb{N}}}$ and
$(m_{k})_{k\in\operatorname{\mathbb{N}}}$ recursively (we set
$n_{0}:=m_{0}:=0$)

$n_{k}:=\min\left\\{n\in\operatorname{\mathbb{N}}:n>n_{k-1},|s_{n}|\leq|t_{n}|\right\\}\qquad
m_{k}:=\min\left\\{n\in\operatorname{\mathbb{N}}:n>m_{k-1},|s_{n}|>|t_{n}|\right\\},\quad\text{for
}k\in\operatorname{\mathbb{N}}.$

Then, by construction, $(n_{k})_{k\in\operatorname{\mathbb{N}}}$ and
$(m_{k})_{k\in\operatorname{\mathbb{N}}}$ form a partition of
$\operatorname{\mathbb{N}}$. In the following, we only consider the case where
both those sequences tend to infinity. The limit in (21) can then be written
as

$\displaystyle\limsup_{n\rightarrow\infty}\left|\mathbb{E}_{\mu_{\alpha,\beta}}\left[e^{it_{n}V(X)+is_{n}W(X)}\right]\right|$
$\displaystyle=\limsup_{k\rightarrow\infty}\max\left\\{\left|\mathbb{E}_{\mu_{\alpha,\beta}}\left[e^{it_{n_{k}}V(X)+is_{n_{k}}W(X)}\right]\right|,\left|\mathbb{E}_{\mu_{\alpha,\beta}}\left[e^{it_{m_{k}}V(X)+is_{m_{k}}W(X)}\right]\right|\right\\}.$

By Proposition 3.8, we find $\varepsilon,\varepsilon^{\prime}\in(0,1)$ and two
sequences $(\delta_{k})_{k\in\operatorname{\mathbb{N}}}$ and
$(\delta^{\prime}_{k})_{k\in\operatorname{\mathbb{N}}}$ with
$\delta_{k}\rightarrow 0,\delta_{k}^{\prime}\rightarrow 0$ such that

$\displaystyle\limsup_{k\rightarrow\infty}\max\left\\{\left|\mathbb{E}_{\mu_{\alpha,\beta}}\left[e^{it_{n_{k}}V(X)+is_{n_{k}}W(X)}\right]\right|,\left|\mathbb{E}_{\mu_{\alpha,\beta}}\left[e^{it_{m_{k}}V(X)+is_{m_{k}}W(X)}\right]\right|\right\\}$
$\displaystyle=\limsup_{k\rightarrow\infty}\max\left\\{1-\varepsilon+\delta_{k},1-\varepsilon^{\prime}+\delta_{k}^{\prime}\right\\}$
$\displaystyle=1-\min\\{\varepsilon,\varepsilon^{\prime}\\}<1.$

This shows the desired result.

∎

###  3.3 Proofs of the Theorems

We start by proving a general lemma needed in several of the upcoming proofs.

######  Lemma 3.11.

Let the assumptions of Lemma 3.10 be satisfied for a pair
$(\alpha,\beta)\in(-\infty,0)\times(0,\infty)$. Then, for the sequence of
distribution functions $(F_{n})_{n\in\operatorname{\mathbb{N}}}$, we have

$\int_{x=0}^{\infty}\int_{y=-\infty}^{0}e^{\sqrt{n}\alpha x+\sqrt{n}\beta
y}dF_{n}(x,y)=\int_{x=0}^{\infty}\int_{y=-\infty}^{0}e^{\sqrt{n}\alpha
x+\sqrt{n}\beta y}dG^{\Sigma}(x,y)+o(n^{-1}).$

If $\beta=0$ in the integral above and $t\in\operatorname{\mathbb{R}}$, then
we get

$\int_{x=0}^{\infty}\int_{y=-\infty}^{t}e^{\sqrt{n}\alpha x+\sqrt{n}\beta
y}dF_{n}(x,y)=\int_{x=0}^{\infty}\int_{y=-\infty}^{t}e^{\sqrt{n}\alpha
x}dG^{\Sigma}(x,y)+O(n^{-1}),$

where the implicit constant in the $O$-term is independent of $t$.

###### Proof.

We start with the first case, where
$(\alpha,\beta)\in(-\infty,0)\times(0,\infty)$. Since $\mu_{\alpha,\beta}$
exists for a pair $(\alpha,\beta)\in(-\infty,0)\times(0,\infty)$, we have that

$\mathbb{E}_{\mu_{\alpha,\beta}}\left[\left|\left|\left(V(X),W(X)\right)\right|\right|_{2}^{2}\right]<\infty,$

and thus we can write

$F_{n}(x,y)=G^{\Sigma}(x,y)+\sum_{j=1}^{2}n^{-j/2}F_{j}(x,y)+R_{n}(x,y),$

with $R_{n}(x,y)=o(n^{-1})$, uniformly in
$(x,y)\in\operatorname{\mathbb{R}}^{2}$ and the density of $F_{j}$ is of the
form $g^{\Sigma}(x,y)\pi_{j}(x,y)$ with $\pi_{j}$ having degree $3j$. This
leads to (recall that $\alpha\in(-\infty,0)$)

$\displaystyle\int_{x=0}^{\infty}\int_{y=-\infty}^{0}e^{\sqrt{n}\alpha
x+\sqrt{n}\beta y}dR_{n}(x,y)$ $\displaystyle\leq
R_{n}\left((-\infty,0)\times(0,\infty)\right)$ $\displaystyle=o(n^{-1}).$

There exist constants $c_{1},c_{2}\in(0,\infty)$ such that, for $j=1,2$, we
have

$\left|g^{\Sigma}(x,y)\pi_{j}(x,y)\right|\leq
c_{1}e^{-c_{2}(x^{2}+y^{2})},\quad\text{for all
}(x,y)\in\operatorname{\mathbb{R}}^{2}.$

From that we infer (we denote $\gamma:=\min\\{|\alpha|,\beta\\}$)

$\displaystyle\int_{x=0}^{\infty}\int_{y=-\infty}^{0}e^{\sqrt{n}\alpha
x+\sqrt{n}\beta y}dF_{j}(x,y)$
$\displaystyle\leq\int_{x=0}^{\infty}\int_{y=-\infty}^{0}e^{\sqrt{n}\alpha
x+\sqrt{n}\beta y}c_{1}e^{-c_{2}(x^{2}+y^{2})}dydx$ $\displaystyle\leq
c_{1}\left(\int_{0}^{\infty}e^{-\sqrt{n}\gamma x}e^{-c_{2}x^{2}}dx\right)^{2}$
$\displaystyle=O\left(n^{-1}\right).$

This implies that

$\sum_{j=1}^{2}\int_{x=0}^{\infty}\int_{y=-\infty}^{0}e^{\sqrt{n}\alpha
x+\sqrt{n}\beta y}d\left(n^{-j/2}F_{j}(x,y)\right)=o(n^{-1}).$

This establishes the first claim. Now let $\alpha\in(-\infty,0)$ and
$\beta=0$. Then, again, we have

$\mathbb{E}_{\mu_{\alpha,0}}\left[\left|\left|\left(V(X),W(X)\right)\right|\right|_{2}^{2}\right]<\infty,$

allowing us to write

$F_{n}(x,y)=G^{\Sigma}(x,y)+n^{-1/2}F_{1}(x,y)+R_{n}(x,y),$

with $R_{n}(x,y)=O(n^{-1})$, uniformly in
$(x,y)\in\operatorname{\mathbb{R}}^{2}$ and the density of $F_{j}$ is of the
form $g^{\Sigma}(x,y)\pi_{1}(x,y)$ with $\pi_{1}$ having degree $3$ and
$g^{\Sigma}$ is the density of a bivariate normal distribution with mean $0$
and covariance matrix $\Sigma\in\operatorname{\mathbb{R}}^{2\times 2}$. As
before, we get

$\displaystyle\int_{x=0}^{\infty}\int_{y=-\infty}^{t}e^{\alpha\sqrt{n}x}dF_{n}(x,y)$
$\displaystyle=\int_{x=0}^{\infty}\int_{y=-\infty}^{t}e^{\alpha\sqrt{n}x}dG^{\Sigma}(x,y)+\int_{x=0}^{\infty}\int_{y=-\infty}^{t}e^{\alpha\sqrt{n}x}d\left(n^{-1/2}F_{1}(x,y)\right)+O(n^{-1}).$

We further find constants $c_{1},c_{2}\in(0,\infty)$ with

$\left|\pi_{1}(x,y)g^{\Sigma}(x,y)\right|\leq c_{1}e^{-c_{2}(x^{2}+y^{2})}.$

This implies

$\displaystyle\int_{x=0}^{\infty}\int_{y=-\infty}^{t}e^{\alpha\sqrt{n}x}d\left(n^{-1/2}F_{1}(x,y)\right)$
$\displaystyle\ll
n^{-1/2}\int_{x=0}^{\infty}e^{\alpha\sqrt{n}x}e^{-c_{2}x^{2}}\underbrace{\int_{y=-\infty}^{t}e^{-c_{2}y^{2}}dy}_{\ll
1}dx$ $\displaystyle\ll n^{-1/2}\int_{x=0}^{\infty}e^{\alpha\sqrt{n}x}dx$
$\displaystyle\ll n^{-1}.$

∎

The next lemma gives an estimate for the asymptotic volume of a high
dimensional Orlicz ball. Such a result was first obtained in [22, Theorem A],
where the error term was of order $o(1)$. For the proof of Theorem B, we need
an error of order $O(n^{-1/2})$.

######  Lemma 3.12.

Let $R\in(0,\infty)$ and $V:\operatorname{\mathbb{R}}\rightarrow[0,\infty)$ be
an Orlicz function. Then the asymptotic volume of $\mathbb{B}^{(n,V)}_{R}$ is
given by

$\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\right)=\frac{e^{-\alpha_{*}nR}Z_{\alpha_{*},0}^{n}}{\sqrt{2\pi
n}\sigma_{*}|\alpha_{*}|}\left(1+O\left(n^{-1/2}\right)\right).$

######  Remark 3.13.

We note that Lemma 3.12 is a special case of [8, Theorem 1.1]. Nevertheless,
in the following, we present an alternative proof, resting on the methods
developed in this article.

###### Proof.

(Lemma 3.12) We have

$\displaystyle\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\right)$
$\displaystyle=\int_{\operatorname{\mathbb{R}}^{n}}\mathbbm{1}_{\mathbb{B}^{(n,V)}_{R}}(x)dx_{1}\ldots
dx_{n}$
$\displaystyle=Z_{\alpha_{*},0}^{n}\int_{\operatorname{\mathbb{R}}^{n}}\mathbbm{1}_{\left[\sum_{i=1}^{n}V(x_{i})\leq
nR\right]}\frac{1}{Z_{\alpha_{*},0}^{n}}e^{(\alpha_{*}-\alpha_{*})\sum_{i=1}^{n}V(x_{i})}dx_{1}\ldots
dx_{n}$
$\displaystyle=Z_{\alpha_{*},0}^{n}e^{-n\alpha_{*}R}\int_{x=-\infty}^{0}\int_{y=-\infty}^{\infty}e^{-\alpha\sqrt{n}x}dF_{n}(x,y),$

where $F_{n}$ is the distribution function of the vector

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left(V(X_{i})-R,W(X_{i})-m\right),$

with $(X_{i})_{i\in\operatorname{\mathbb{N}}}$ being an iid sequence all
distributed according to $\mu_{\alpha_{*},0}$. By Lemma 3.10 together with
Lemma 3.11, we get

$\displaystyle\int_{x=-\infty}^{0}\int_{y=-\infty}^{\infty}e^{-\alpha\sqrt{n}x}dF_{n}(x,y)$
$\displaystyle=\int_{x=-\infty}^{0}\int_{y=-\infty}^{\infty}e^{-\alpha\sqrt{n}x}dG^{\Sigma}(x,y)+O(n^{-1})$
$\displaystyle=\frac{1}{\sqrt{2\pi}\sigma_{*}}\int_{x=0}^{\infty}e^{\alpha_{*}\sqrt{n}x}e^{-\frac{x^{2}}{2\sigma_{*}}}dx+O(n^{-1})$
$\displaystyle=\frac{1}{\sqrt{2\pi
n}\sigma_{*}}\int_{x=0}^{\infty}e^{\alpha_{*}x}e^{-\frac{x^{2}}{2n\sigma_{*}}}dx+O(n^{-1})$
$\displaystyle=\frac{1}{\sqrt{2\pi
n}\sigma_{*}}\int_{x=0}^{\infty}e^{\alpha_{*}x}dx+\frac{1}{\sqrt{2\pi
n}\sigma_{*}}\int_{x=0}^{\infty}e^{\alpha_{*}x}\left(e^{-\frac{x^{2}}{2n\sigma_{*}}}-1\right)dx+O(n^{-1})$
$\displaystyle=\frac{1}{\sqrt{2\pi
n}\sigma_{*}|\alpha_{*}|}+\frac{1}{\sqrt{2\pi
n}\sigma_{*}}\int_{x=0}^{\infty}e^{\alpha_{*}x}\left(e^{-\frac{x^{2}}{2n\sigma_{*}}}-1\right)dx+O(n^{-1}).$

We take a closer look at the second integral in the expression above, where we
get (we use the estimate $|e^{t}-1|\leq|t|\max\\{e^{t},1\\}$ for
$t\in\operatorname{\mathbb{R}}$)

$\displaystyle\frac{1}{\sqrt{2\pi
n}\sigma_{*}}\left|\int_{x=0}^{\infty}e^{\alpha_{*}x}\left(e^{-\frac{x^{2}}{2n\sigma_{*}}}-1\right)dx\right|$
$\displaystyle\ll\frac{1}{n^{3/2}}\int_{x=0}^{\infty}x^{2}e^{\alpha_{*}x}\underbrace{\max\left\\{1,e^{-\frac{x^{2}}{2n\sigma_{*}}}\right\\}}_{\leq
1}dx$ $\displaystyle\ll\frac{1}{n^{3/2}}.$

This finishes the proof. ∎

###### Proof.

(Theorem A) We only prove $(i)$, since the proof of $(ii)$ can be carried out
in a similar way. By Lemma 3.3, there exists an $\varepsilon\in(0,\infty)$
such that, for all $t\in(m,m+\varepsilon)$, we find a pair
$(\alpha,\beta)\in(-\infty,0)\times(0,\infty)$ with

$\left(\begin{array}[]{c}R\\\
t\end{array}\right)=\left(\begin{array}[]{c}\mathbb{E}_{\mu_{\alpha,\beta}}\left[V(X)\right]\\\
\mathbb{E}_{\mu_{\alpha,\beta}}\left[W(X)\right]\end{array}\right).$

Let $(X_{i})_{i\in\operatorname{\mathbb{N}}}$ be an iid sequnce of random
variables with $X_{1}\sim\mu_{\alpha,\beta}$. Then we can introduce the
following sequence of centered random variables

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left(Y_{i},Z_{i}\right):=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left(V(X_{i})-R,W(X_{i})-t\right),\quad
n\in\operatorname{\mathbb{N}}.$

We denote the distribution function of this sequence by $F_{n}$ and recall
that $F_{n}$ has an Edgeworth expansion by Lemma 3.10. We now get

$\displaystyle\operatorname{vol}_{n}\left(\mathbb{B}^{(n,V)}_{R}\cap\left(\mathbb{B}^{(n,W)}_{t}\right)^{c}\right)$
$\displaystyle=\int_{\operatorname{\mathbb{R}}^{n}}\mathbbm{1}_{\mathbb{B}^{(n,V)}_{R}}(x)\mathbbm{1}_{\left(\mathbb{B}^{(n,W)}_{t}\right)^{c}}(x)dx_{1}\ldots
dx_{n}$
$\displaystyle=Z_{\alpha,\beta}^{n}\int_{\operatorname{\mathbb{R}}^{n}}\mathbbm{1}_{\left[\sum_{i=1}^{n}V(x_{i})\leq
nR\right]}\mathbbm{1}_{\left[\sum_{i=1}^{n}W(x_{i})>nt\right]}\frac{1}{Z_{\alpha,\beta}^{n}}e^{(\alpha-\alpha)\sum_{i=1}^{n}V(x_{i})+(\beta-\beta)\sum_{i=1}^{n}W(x_{i})}dx_{1}\ldots
dx_{n}$ $\displaystyle=Z_{\alpha,\beta}^{n}e^{-n\alpha R-n\beta
t}\mathbb{E}\left[\mathbb{1}_{\left[\frac{1}{\sqrt{n}}\sum_{i=1}^{n}Y_{i}\leq
0\right]}\mathbb{1}_{\left[\frac{1}{\sqrt{n}}\sum_{i=1}^{n}Z_{i}>0\right]}e^{\sqrt{n}\alpha\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(Y_{i}-R)+\sqrt{n}\beta\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(Z_{i}-t)}\right]$
$\displaystyle=Z_{\alpha,\beta}^{n}e^{-n\alpha R-n\beta
t}\int_{x=0}^{\infty}\int_{y=-\infty}^{0}e^{-\sqrt{n}\beta x-\sqrt{n}\alpha
y}dF_{n}(x,y).$

By Lemma 3.10, we can write

$F_{n}(x,y)=G^{\Sigma}(x,y)+\sum_{j=1}^{2}n^{-j/2}F_{j}(x,y)+R_{n}(x,y),$

where $G^{\Sigma}$ is the distribution function of a normal distribution with
mean $0$ and variance matrix $\Sigma\in\operatorname{\mathbb{R}}^{2\times 2}$
and $R_{n}(x,y)=o(n^{-1})$, uniformly in
$(x,y)\in\operatorname{\mathbb{R}}^{2}$. By Lemma 3.11 we get

$\displaystyle\int_{x=0}^{\infty}\int_{y=-\infty}^{0}e^{-\sqrt{n}\beta
x-\sqrt{n}\alpha y}dF_{n}(x,y)$
$\displaystyle=\int_{x=0}^{\infty}\int_{y=0}^{\infty}e^{-\sqrt{n}\beta
x+\sqrt{n}\alpha y}dG^{\Sigma}(x,y)+o(n^{-1})$
$\displaystyle=\int_{x=0}^{\infty}\int_{y=0}^{\infty}e^{-\sqrt{n}(\beta
x-\alpha
y)}\frac{1}{2\pi|\Sigma|^{1/2}}e^{-\frac{1}{2}(x,y)\Sigma^{-1}\left(\begin{smallmatrix}x\\\
y\end{smallmatrix}\right)}dxdy+o(n^{-1})$ $\displaystyle=\frac{1}{2\pi
n|\Sigma|^{1/2}}\int_{x=0}^{\infty}\int_{y=0}^{\infty}e^{-\beta x+\alpha
y}\underbrace{e^{-\frac{1}{2n}(x,y)\Sigma^{-1}\left(\begin{smallmatrix}x\\\
y\end{smallmatrix}\right)}}_{=:f_{n}(x,y)}dxdy+o(n^{-1}).$

We now observe that $f_{n}(x,y)\leq 1$ for all
$(x,y)\in\operatorname{\mathbb{R}}^{2}$ and for all
$n\in\operatorname{\mathbb{N}}$. Since $e^{-\beta x+\alpha y}$ is integrable,
the dominated convergence theorem yields

$\displaystyle\frac{1}{2\pi
n|\Sigma|^{1/2}}\int_{x=0}^{\infty}\int_{y=0}^{\infty}e^{-\beta x+\alpha
y}f_{n}(x,y)dxdy$ $\displaystyle=\frac{1}{2\pi
n|\Sigma|^{1/2}}\int_{x=0}^{\infty}\int_{y=0}^{\infty}e^{-\beta x+\alpha
y}dxdy+o(n^{-1})$ $\displaystyle=\frac{1}{2\pi
n(-\alpha)\beta|\Sigma|^{1/2}}+o(n^{-1}).$

By Lemma 3.12, it holds that

$\operatorname{vol}_{n}\left(\mathbb{B}^{(n,V)}_{R}\right)=e^{-\alpha^{*}nR}Z_{\alpha^{*},0}^{n}\frac{1}{\sqrt{2\pi
n}\sigma_{*}|\alpha_{*}|}\left(1+O\left(n^{-1/2}\right)\right),$

where $\sigma_{*}^{2}=\mathbb{V}_{\mu_{\alpha_{*},0}}\left[V(X)\right]$. This
implies that

$\frac{\operatorname{vol}_{n}\left(\mathbb{B}^{(n,V)}_{R}\cap\mathbb{B}^{(n,W)}_{t}\right)}{\operatorname{vol}_{n}\left(\mathbb{B}^{(n,V)}_{R}\right)}=1-e^{-n\left(\varphi(\alpha^{*},0)-\varphi(\alpha,\beta)+(\alpha-\alpha^{*})R+\beta
t\right)}\frac{|\alpha_{*}|}{\sqrt{2\pi
n}|\alpha|\beta\bar{\sigma}}\left(1+o(1)\right),$

where

$\bar{\sigma}^{2}=\frac{\sigma_{*}^{2}}{|\Sigma|}$

is the second entry in the diagonal of $\Sigma^{-1}$. Since, by construction,

$\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)=\left(\nabla\varphi\right)^{-1}\left(\begin{array}[]{c}R\\\
t\end{array}\right)\quad\text{and}\quad\left(\begin{array}[]{c}\alpha^{*}\\\
0\end{array}\right)=\left(\nabla\varphi\right)^{-1}\left(\begin{array}[]{c}R\\\
m\end{array}\right),$

we recover the definition of $\operatorname{\mathbb{I}}$ and thus the estimate
in (7).

∎

###### Proof.

(Theorem C) We prove a more general result, where instead of the
$\ell_{2}$-norm we work with another Orlicz function $W$, such that $V$ and
$W$ satisfy Assumption A. We define $\varepsilon\in(0,\infty)$ to be the
largest constant such that, for any $\delta\in(0,\varepsilon)$, we find pairs
$(\alpha_{1},\beta_{1})\in(-\infty,0)\times(0,\infty)$ and
$(\alpha_{2},\beta_{2})\in(-\infty,0)^{2}$ with

$\nabla\varphi(\alpha_{1},\beta_{1})=\left(\begin{array}[]{c}R\\\
m+\delta\end{array}\right)\quad\text{and}\quad\nabla\varphi(\alpha_{2},\beta_{2})=\left(\begin{array}[]{c}R\\\
m-\delta\end{array}\right).$ (22)

This is possible by Lemma 3.3 and Lemma 3.5. We now obtain

$\displaystyle\mathbb{P}\left[\left|\frac{1}{n}\sum_{i=1}^{n}W\left(X_{i}^{(n,V)}\right)-m\right|\geq\delta\right]$
$\displaystyle=\mathbb{P}\left[\frac{1}{n}\sum_{i=1}^{n}W\left(X_{i}^{(n,V)}\right)\geq
m+\delta\right]+\mathbb{P}\left[\frac{1}{n}\sum_{i=1}^{n}W\left(X_{i}^{(n,V)}\right)\leq
m-\delta\right].$

Analogously to the proof of Theorem A, we now get

$\mathbb{P}\left[\left|\frac{1}{n}\sum_{i=1}^{n}W\left(X_{i}^{(n,V)}\right)-m\right|\geq\delta\right]=\frac{1}{\sqrt{2\pi
n}\sigma_{1}}e^{-n\mathbb{I}(R,m+\delta)}(1+o(1))+\frac{1}{\sqrt{2\pi
n}\sigma_{2}}e^{-n\mathbb{I}(R,m-\delta)}(1+o(1)),$

where $\mathbb{I}$ is the same GRF as in Theorem A and

$\sigma_{i}^{2}:=\mathbb{V}_{\mu_{\alpha_{i},\beta_{i}}}\left[V(X)\right],\quad
i=1,2.$

The claim now follows by defining

$\tilde{\sigma}:=\begin{cases}\sigma_{1},&\text{if}\quad\mathbb{I}(R,m+\delta)>\mathbb{I}(R,m-\delta)\\\
\sigma_{2},&\text{if}\quad\mathbb{I}(R,m+\delta)<\mathbb{I}(R,m-\delta)\\\
\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}},&\text{if}\quad\mathbb{I}(R,m+\delta)=\mathbb{I}(R,m-\delta).\end{cases}$

∎

###### Proof.

(Theorem B) We start with the proof of the Berry–Esseen result, where we
recall that there exists an $\alpha_{*}\in(-\infty,0)$ such that

$\left(\begin{array}[]{c}\mathbb{E}_{\mu_{\alpha_{*},0}}[V(X)]\\\
\mathbb{E}_{\mu_{\alpha_{*},0}}[W(X)]\end{array}\right)=\left(\begin{array}[]{c}R\\\
m\end{array}\right).$

This leads to

$\displaystyle\mathbb{P}\left[\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[W\left(X_{i}^{(n,V)}\right)-m\right]\leq
t\right]$
$\displaystyle=\frac{1}{\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\right)}\int_{\operatorname{\mathbb{R}}^{n}}\mathbb{1}_{\mathbb{B}_{R}^{(n,V)}}(x)\mathbb{1}_{\left[\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left(W(x_{i})-m\right)\leq
t\right]}dx_{1}\ldots dx_{n}$
$\displaystyle=\frac{Z_{\alpha_{*}}^{-n}e^{n\alpha_{*}R}}{\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\right)}\int_{\operatorname{\mathbb{R}}^{n}}\mathbb{1}_{\left[\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[V(x_{i})-R\right]\leq
0\right]}\mathbb{1}_{\left[\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[W(x_{i})-m\right]\leq
t\right]}e^{\alpha_{*}\sum_{i=1}^{n}\left[V(x_{i})-R\right]}dx_{1}\ldots
dx_{n}$
$\displaystyle=\frac{Z_{\alpha_{*}}^{-n}e^{n\alpha_{*}R}}{\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\right)}\int_{x=-\infty}^{0}\int_{y=-\infty}^{t}e^{\sqrt{n}\alpha_{*}x}dF_{n}(x,y),$

where $F_{n}$ is the distribution function of the vector

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left(V(X_{i})-R,W(X_{i})-m\right),$

with $(X_{i})_{i\in\operatorname{\mathbb{N}}}$ being an iid sequence with
$X_{1}$ being distributed according to $\mu_{\alpha_{*},0}$. By Lemma 3.12, it
holds that

$\operatorname{vol}_{n}\left(\mathbb{B}^{(n,V)}_{R}\right)=e^{-\alpha^{*}nR}Z_{\alpha^{*},0}^{n}\frac{1}{\sqrt{2\pi
n}\sigma_{*}|\alpha_{*}|}(1+O(n^{-1/2})).$

Together with Lemma 3.11 we now get (let $\Sigma=(\sigma_{i,j})_{i,j=1,2}$ and
$\Sigma^{-1}=(\rho_{i,j})_{i,j=1,2}$ and recall the identity
$\rho_{2,2}=\frac{\sigma_{2,2}}{|\Sigma|}$)

$\begin{split}\mathbb{P}\left[\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[W\left(X_{i}^{(n,V)}\right)-m\right]\leq
t\right]&=\frac{\sqrt{n}|\alpha_{*}|\sigma_{*}}{\sqrt{2\pi}|\Sigma|^{1/2}}\int_{x=0}^{\infty}\int_{y=-\infty}^{t}e^{-\sqrt{n}\alpha_{*}x}e^{-(x,y)\Sigma^{-1}\left(\begin{smallmatrix}x\\\
y\end{smallmatrix}\right)}dxdy+O(n^{-1/2})\\\
&=\frac{\sqrt{\rho_{2,2}}}{\sqrt{2\pi}}\int_{x=0}^{\infty}\int_{y=-\infty}^{t}e^{-x}e^{-\left(\frac{x}{\sqrt{n}},y\right)\Sigma^{-1}\left(\begin{smallmatrix}\frac{x}{\sqrt{n}}\\\
y\end{smallmatrix}\right)}dydx+O(n^{-1/2}).\end{split}$ (23)

We take a closer look at the integral above, where we get

$\displaystyle\int_{x=0}^{\infty}\int_{y=-\infty}^{t}e^{-x}e^{-\left(\frac{x}{\sqrt{n}},y\right)\Sigma^{-1}\left(\begin{smallmatrix}\frac{x}{\sqrt{n}}\\\
y\end{smallmatrix}\right)}dydx$
$\displaystyle=\int_{x=0}^{\sqrt{n}}e^{-x}\int_{y=-\infty}^{t}e^{-\left(\frac{x}{\sqrt{n}},y\right)\Sigma^{-1}\left(\begin{smallmatrix}\frac{x}{\sqrt{n}}\\\
y\end{smallmatrix}\right)}dydx+\int_{x=\sqrt{n}}^{\infty}e^{-x}\int_{y=-\infty}^{t}e^{-\left(\frac{x}{\sqrt{n}},y\right)\Sigma^{-1}\left(\begin{smallmatrix}\frac{x}{\sqrt{n}}\\\
y\end{smallmatrix}\right)}dydx$
$\displaystyle=\int_{x=0}^{\sqrt{n}}e^{-x}\int_{y=-\infty}^{t}\exp\left(-\frac{x^{2}\rho_{1,1}}{2n}-\frac{xy\rho_{2,1}}{\sqrt{n}}-\frac{y^{2}\rho_{2,2}}{2}\right)dydx+O\left(n^{-1/2}\right)$
$\displaystyle\qquad\pm\int_{x=0}^{\sqrt{n}}e^{-x}\int_{y=-\infty}^{t}\exp\left(-\frac{y^{2}\rho_{2,2}}{2}\right)dydx$
$\displaystyle=\int_{x=0}^{\sqrt{n}}e^{-x}\int_{y=-\infty}^{t}\exp\left(-\frac{x^{2}\rho_{1,1}}{2n}-\frac{xy\rho_{2,1}}{\sqrt{n}}-\frac{y^{2}\rho_{2,2}}{2}\right)-\exp\left(-\frac{y^{2}\rho_{2,2}}{2}\right)dydx$
$\displaystyle\qquad+\int_{x=0}^{\sqrt{n}}e^{-x}\int_{y=-\infty}^{t}\exp\left(-\frac{y^{2}\rho_{2,2}}{2}\right)dydx+O\left(n^{-1/2}\right).$

We aim to show that the first integral in the expression above is of order
$O(n^{-1/2})$. We recall the trivial estimate

$\left|e^{t}-1\right|\leq|t|\left(1+e^{t}\right),\quad
t\in\operatorname{\mathbb{R}}.$

This yields (we use that
$\left|\frac{x^{2}\rho_{1,1}}{n}+\frac{xy\rho_{2,1}}{\sqrt{n}}\right|\ll\frac{x}{\sqrt{n}}(|y|+1)$)

$\displaystyle\left|\int_{x=0}^{\sqrt{n}}e^{-x}\int_{y=-\infty}^{t}e^{-\frac{y^{2}\rho_{2,2}}{2}}\left[\exp\left(-\frac{x^{2}\rho_{1,1}}{2n}-\frac{xy\rho_{2,1}}{\sqrt{n}}\right)-1\right]dydx\right|$
$\displaystyle\ll\frac{1}{\sqrt{n}}\int_{x=0}^{\sqrt{n}}xe^{-x}\underbrace{\int_{y=-\infty}^{t}(|y|+1)e^{-\frac{y^{2}\rho_{2,2}}{2}-\frac{xy}{\sqrt{n}}}dy}_{\ll
1}dx$ $\displaystyle\ll\frac{1}{\sqrt{n}}\int_{x=0}^{\sqrt{n}}xe^{-x}dx$
$\displaystyle\ll\frac{1}{\sqrt{n}}.$

From that we infer

$\displaystyle\int_{x=0}^{\infty}\int_{y=-\infty}^{t}e^{-x}e^{-\left(\frac{x}{\sqrt{n}},y\right)\Sigma^{-1}\left(\begin{smallmatrix}\frac{x}{\sqrt{n}}\\\
y\end{smallmatrix}\right)}dydx$
$\displaystyle=\int_{x=0}^{\infty}e^{-x}\int_{y=-\infty}^{t}e^{-\frac{y^{2}\rho_{2,2}}{2}}dydx+O\left(n^{-1/2}\right)$
$\displaystyle=\int_{y=-\infty}^{t}e^{-\frac{y^{2}\rho_{2,2}}{2}}dy+O\left(n^{-1/2}\right).$

Combining this estimate with Equation (23) gives us

$\displaystyle\mathbb{P}\left[\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[W\left(X_{i}^{(n,V)}\right)-m\right]\leq
t\right]$ $\displaystyle=\phi(t)+O\left(n^{-1/2}\right),$

where $\phi$ is the cumulative distribution function of a normal distribution
with mean $0$ and variance $\rho_{2,2}^{-1}$.

Next we prove the strong law of large numbers using the convergence
Borel–Cantelli lemma. To that end, let $\varepsilon\in(0,\infty)$, where we
will show that

$\sum_{n=1}^{\infty}\mathbb{P}\left[\left|\frac{1}{n}\sum_{i=1}^{n}W\left(X_{i}^{(n,V)}\right)-m\right|>\varepsilon\right]<\infty.$

Let $n\in\operatorname{\mathbb{N}}$ be large, where we have (using an
analogous argument as in the first part of this proof)

$\displaystyle\mathbb{P}\left[\left|\frac{1}{n}\sum_{i=1}^{n}W\left(X_{i}^{(n,V)}\right)-m\right|>\varepsilon\right]$
$\displaystyle=\int_{x=0}^{\infty}\int_{|y|\geq\sqrt{n}\varepsilon}e^{-\alpha_{*}\sqrt{n}x}dF_{n}(x,y)\left(1+O\left(n^{-1/2}\right)\right)$
$\displaystyle\ll\int_{x=0}^{\infty}\int_{|y|\geq\sqrt{n}\varepsilon}e^{-\alpha_{*}\sqrt{n}x}dF_{n}(x,y).$

Since

$\mathbb{E}_{\mu_{\alpha_{*},0}}\left[\left|\left|\left(V(X),W(X)\right)\right|\right|_{2}^{3}\right]<\infty,$

we can write

$F_{n}(x,y)=G^{\Sigma}(x,y)+\sum_{j=1}^{3}n^{-j/2}F_{j}(x,y)+R_{n}(x,y),$

with $R_{n}(x,y)=o(n^{-3/2})$, uniformly in
$(x,y)\in\operatorname{\mathbb{R}}^{2}$ and the density of $F_{j}$ is of the
form $g^{\Sigma}(x,y)\pi_{j}(x,y)$ with $\pi_{j}$ having degree $3j$. This
implies

$\displaystyle\int_{x=0}^{\infty}\int_{|y|\geq\sqrt{n}\varepsilon}e^{-\alpha_{*}\sqrt{n}x}dF_{n}(x,y)$
$\displaystyle=\int_{x=0}^{\infty}\int_{|y|\geq\sqrt{n}\varepsilon}e^{-\alpha_{*}\sqrt{n}x}dG^{\Sigma}(x,y)+\sum_{j=1}^{3}n^{-j/2}\int_{x=0}^{\infty}\int_{|y|\geq\sqrt{n}\varepsilon}e^{-\alpha_{*}\sqrt{n}x}dF_{j}(x,y)+o(n^{-3/2})$
$\displaystyle=\int_{x=0}^{\infty}\int_{|y|\geq\sqrt{n}\varepsilon}e^{-\alpha_{*}\sqrt{n}x}dG^{\Sigma}(x,y)+o(n^{-3/2})$
$\displaystyle\ll\int_{x=0}^{\infty}e^{-\alpha_{*}\sqrt{n}x}\int_{|y|\geq\sqrt{n}\varepsilon}e^{-c(x^{2}+y^{2})}dydx+o(n^{-3/2})$
$\displaystyle\ll\int_{x=0}^{\infty}e^{-\alpha_{*}\sqrt{n}x-cx^{2}}e^{-c^{\prime}n}dx+o(n^{-3/2})$
$\displaystyle=o(n^{-3/2}),$

where, in the second last line, we used that

$\int_{|y|\geq\sqrt{n}\varepsilon}e^{-cy^{2}}dy\ll e^{-c^{\prime}n},$

for some constant $c^{\prime}\in(0,\infty)$. This implies that

$\sum_{n=1}^{\infty}\mathbb{P}\left[\left|\frac{1}{n}\sum_{i=1}^{n}W\left(X_{i}^{(n,V)}\right)-m\right|>\varepsilon\right]=\sum_{n=1}^{\infty}o(n^{-3/2})<\infty.$

∎

###### Proof.

(of Corollary 1.4) We recall that

$\displaystyle\frac{\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\cap\mathbb{B}_{m}^{(n,W)}\right)}{\operatorname{vol}_{n}\left(\mathbb{B}_{R}^{(n,V)}\right)}$
$\displaystyle=\mathbb{P}\left[\frac{1}{n}\sum_{i=1}^{n}W\left(X_{i}^{(n,V)}\right)\leq
m\right]$
$\displaystyle=\mathbb{P}\left[\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[W\left(X_{i}^{(n,V)}\right)-m\right]\leq
0\right],$

where $X^{(n,V)}$ is uniformly distributed on $\mathbb{B}_{R}^{(n,V)}$. The
last probability in the expression above converges to $\frac{1}{2}$ by Theorem
B, which proves the claim. ∎

### Acknowledgement

LF and JP are supported by the Austrian Science Fund (FWF) Project P32405
Asymptotic geometric analysis and applications. LF is also supported by the
FWF Project P 35322 Zufall und Determinismus in Analysis und Zahlentheorie.

## References

  * [1] D. Alonso-Gutiérrez and J. Bastero.  The variance conjecture on some polytopes.  Asymptotic Geometric Analysis: Proceedings of the Fall 2010 Fields Institute Thematic Program, pages 1–20, 2013.
  * [2] D. Alonso-Gutiérrez and J. Bastero.  The variance conjecture on hyperplane projections of the $\ell_{p}^{n}$ balls.  Rev. Mat. Iberoam, 34(2):879–904, 2018.
  * [3] D. Alonso-Gutiérrez and J. Prochno.  Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures.  J. Funct. Anal., 282(1), 2022.
  * [4] D. Alonso-Gutiérrez, J. Prochno, and C. Thaele.  Large deviations, moderate deviations, and the kls conjecture.  J. Funct. Anal., 280(1):108779, 2021.
  * [5] D. Alonso-Gutiérrez, J. Prochno, and C. Thäle.  Large deviations for high-dimensional random projections of $\ell_{p}^{n}$-balls.  Adv. Appl. Math., 99:1–35, 2018.
  * [6] M. Anttila, K. Ball, and I. Perissinaki.  The central limit problem for convex bodies.  Trans. Am. Math. Soc., 355(12):4723–4735, 2003.
  * [7] R. R. Bahadur and R. Ranga Rao.  On deviations of the sample mean.  Ann. Math. Statist., 31:1015–1027, 1960.
  * [8] F. Barthe and P. Wolff.  Volume properties of high-dimensional Orlicz balls.  High Dimensional Probability IX: The Ethereal Volume, pages 75–95, 2023.
  * [9] R. N. Bhattacharya and R. Ranga Rao.  Normal approximation and asymptotic expansions.  SIAM, 2010.
  * [10] S. G. Bobkov and A. Koldobsky.  On the central limit property of convex bodies.  In Geometric Aspects of Functional Analysis: Israel Seminar 2001-2002, pages 44–52. Springer, 2003.
  * [11] J. Bourgain.  On the distribution of polynomials on high-dimensional convex sets.  In Geometric aspects of functional analysis (1989–90), volume 1469 of Lecture Notes in Math., pages 127–137. Springer, Berlin, 1991.
  * [12] J. Bourgain.  On the isotropy-constant problem for “PSI-2”-bodies.  In Geometric aspects of functional analysis, volume 1807 of Lecture Notes in Math., pages 114–121. Springer, Berlin, 2003.
  * [13] Y. Chen.  An almost constant lower bound of the isoperimetric coefficient in the KLS conjecture.  Geom. Funct. Anal., 31(1):34–61, 2021.
  * [14] P. Diaconis and D. Freedman.  Asymptotics of graphical projection pursuit.  Ann. Stat., pages 793–815, 1984.
  * [15] P. Eichelsbacher and M. Löwe.  Moderate deviations for iid random variables.  ESAIM: Probab. Stat., 7:209–218, 2003.
  * [16] L. Frühwirth and J. Prochno.  Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls.  J. Math. Anal. Appl., 536(1):128169, 2024.
  * [17] N. Gantert, S. S. Kim, and K. Ramanan.  Cramér’s theorem is atypical.  In Advances in the Mathematical Sciences: Research from the 2015 Association for Women in Mathematics Symposium, pages 253–270. Springer, 2016\.
  * [18] N. Gantert, S. S. Kim, and K. Ramanan.  Large deviations for random projections of $\ell^{p}$ balls.  Ann. Probab., 45(6B):4419–4476, 2017.
  * [19] O. Guédon and E. Milman.  Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures.  Geom. Funct. Anal., 21(5):1043–1068, 2011.
  * [20] C. Hao, A. Kamińska, and N. Tomczak-Jaegermann.  Orlicz spaces with convexity or concavity constant one.  J. Math. Anal. Appl., 320(1):303–321, 2006.
  * [21] S. G. G. Johnston and J. Prochno.  A Maxwell principle for generalized Orlicz balls.  Ann. Inst. H. Poincaré Probab. Statist., 59(3):1223–1247, 2023\.
  * [22] Z. Kabluchko and J. Prochno.  The maximum entropy principle and volumetric properties of Orlicz balls.  J. Math. Anal. Appl., 495(1):124687, 2021.
  * [23] Z. Kabluchko, J. Prochno, and C. Thaele.  A new look at random projections of the cube and general product measures.  Bernoulli, 27(3):2117–2138, 2021.
  * [24] Z. Kabluchko, J. Prochno, and C. Thäle.  High-dimensional limit theorems for random vectors in $\ell_{p}^{n}$-balls.  Commun. Contemp. Math., 21(1):1750092, 30, 2019.
  * [25] Z. Kabluchko, J. Prochno, and C. Thäle.  High-dimensional limit theorems for random vectors in $\ell_{p}^{n}$-balls. ii.  Commun. Contemp. Math., 23(03):1950073, 2021.
  * [26] A. Kamińska.  The criteria for local uniform rotundity of Orlicz spaces.  Studia Math., 79(3):201–215, 1984.
  * [27] T. Kaufmann.  Sharp asymptotics for q-norms of random vectors in high-dimensional $\ell_{p}^{n}$-balls.  Mod. Stoch.: Theory Appl., 8(2):239–274, 2021.
  * [28] S. S. Kim, Y. Liao, and K. Ramanan.  An asymptotic thin shell condition and large deviations for random multidimensional projections.  Adv. Appl. Math., 134:102306, 2022.
  * [29] S. S. Kim and K. Ramanan.  A conditional limit theorem for high-dimensional $\ell^{p}$-spheres.  J. Appl. Probab., 55(4):1060–1077, 2018.
  * [30] B. Klartag.  A central limit theorem for convex sets.  Invent. Math., 168(1):91–131, 2007.
  * [31] B. Klartag.  Logarithmic bounds for isoperimetry and slices of convex sets.  Ars Inven. Anal., pages Paper No. 4, 17, 2023.
  * [32] B. Klartag and J. Lehec.  Bourgain’s slicing problem and KLS isoperimetry up to polylog.  Geom. Funct. Anal., 32(5):1134–1159, 2022.
  * [33] B. Klartag and J. Lehec.  Isoperimetric inequalities in high-dimensional convex sets.  arXiv preprint arXiv:2406.01324, 2024.
  * [34] A. V. Kolesnikov and E. Milman.  The KLS isoperimetric conjecture for generalized Orlicz balls.  Ann. Probab., 46(6):3578–3615, 2018.
  * [35] P. Kosmol and D. Müller-Wichards.  Optimization in function spaces: with stability considerations in Orlicz spaces, volume 13.  Walter de Gruyter, 2011.
  * [36] S. Kwapień and C. Schütt.  Some combinatorial and probabilistic inequalities and their application to Banach space theory.  Studia Math., 82(1):91–106, 1985.
  * [37] Y. T. Lee and S. S. Vempala.  Stochastic localization+ Stieltjes barrier= tight bound for log-Sobolev.  In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 1122–1129, 2018.
  * [38] Y. Liao and K. Ramanan.  Geometric sharp large deviations for random projections of $\ell_{p}^{n}$ spheres and balls.  Electronic Journal of Probability, 29:1–56, 2024.
  * [39] A. Naor.  The surface measure and cone measure on the sphere of $\ell_{p}$.  Trans. Am. Math. Soc., 359(3):1045–1079, 2007.
  * [40] V. V. Petrov.  On the probabilities of large deviations for sums of independent random variables.  Teor. Verojatnost. i Primenen, pages 310–322, 1965.
  * [41] J. Prochno.  The large and moderate deviations approach in geometric functional analysis.  arXiv preprint arXiv: 2403.03940, 2024.
  * [42] J. Prochno and C. Schütt.  Combinatorial inequalities and subspaces of $L_{1}$.  Studia Math., 211(1):21–39, 2012.
  * [43] J. Prochno, C. Thäle, and N. Turchi.  Geometry of $\ell_{p}^{n}$-balls: Classical results and recent developments.  High Dimensional Probability VIII: The Oaxaca Volume, pages 121–150, 2019.
  * [44] S. T. Rachev and L. Rüschendorf.  Approximate independence of distributions on spheres and their stability properties.  Ann. Probab., 19(3):1311–1337, 1991.
  * [45] Yves Raynaud and Carsten Schütt.  Some results on symmetric subspaces of $L_{1}$.  Stud. Math., 89(1):27–35, 1988.
  * [46] G. Schechtman and M. Schmuckenschläger.  Another remark on the volume of the intersection of two $L^{n}_{p}$ balls.  In Geometric aspects of functional analysis (1989–90), volume 1469 of Lecture Notes in Math., pages 174–178. Springer, Berlin, 1991.
  * [47] G. Schechtman and J. Zinn.  On the volume of the intersection of two $L^{n}_{p}$ balls.  Proc. Amer. Math. Soc., 110(1):217–224, 1990.
  * [48] G. Schechtman and J. Zinn.  Concentration on the $\ell_{p}^{n}$ ball.  In Geometric Aspects of Functional Analysis: Israel Seminar 1996–2000, pages 245–256. Springer, 2000.
  * [49] M. Schmuckenschläger.  CLT and the Volume of Intersections of $\ell_{p}^{n}$-Balls.  Geometriae Dedicata, 85:189–195, 2001.
  * [50] C. Schütt.  On the embedding of $2$-concave Orlicz spaces into $L^{1}$.  Studia Math., 113(1):73–80, 1995.
  * [51] V. N. Sudakov.  Typical distributions of linear functionals in finite-dimensional spaces of higher dimension.  Doklady Akademii Nauk, 243(6):1402–1405, 1978.
  * [52] H. von Weizsäcker.  Sudakov’s typical marginals, random linear functionals and a conditional central limit theorem.  Probab. Theory Relat. Fields, 107(3):313–324, 1997.
  * [53] J. O. Wojtaszczyk.  The square negative correlation property for generalized Orlicz balls.  In Geometric Aspects of Functional Analysis: Israel Seminar 2004–2005, pages 305–313. Springer, 2007.

Lorenz Frühwirth: Faculty of Computer Science and Mathematics, University of
Passau, Dr.-Hans-Kapfinger-Straße 30, 94032 Passau, Germany.

E-mail<EMAIL_ADDRESS>
Joscha Prochno: Faculty of Computer Science and Mathematics, University of
Passau, Dr.-Hans-Kapfinger-Straße 30, 94032 Passau, Germany.

E-mail<EMAIL_ADDRESS>