# The large and moderate deviations approach in
geometric functional analysis

Joscha Prochno

###### Abstract

The work of Gantert, Kim, and Ramanan [Large deviations for random projections
of $\ell^{p}$ balls, Ann. Probab. 45 (6B), 2017] has initiated and inspired a
new direction of research in the asymptotic theory of geometric functional
analysis. The moderate deviations perspective, describing the asymptotic
behavior between the scale of a central limit theorem and a large deviations
principle, was later added by Kabluchko, Prochno, and Thäle in [High-
dimensional limit theorems for random vectors in $\ell_{p}^{n}$ balls. II,
Commun. Contemp. Math. 23(3), 2021]. These two approaches nicely complement
the classical study of central limit phenomena or non-asymptotic concentration
bounds for high-dimensional random geometric quantities. Beyond studying large
and moderate deviations principles for random geometric quantities that appear
in geometric functional analysis, other ideas emerged from the theory of large
deviations and the closely related field of statistical mechanics, and have
provided new insight and become a source for new developments. Within less
than a decade, a variety of results have appeared and formed this direction of
research. Among others, a connection to the famous Kannan–Lovász–Simonovits
conjecture and the study of moderate and large deviations for isotropic log-
concave random vectors was discovered. In this manuscript, we introduce the
basic principles, survey the work that has been done, and aim to manifest this
direction of research, at the same time making it more accessible to a wider
community of researchers. Keywords. Cramér’s theorem, Gibbs measure, moderate
deviations principle, large deviations principle, Orlicz ball, random
projection, Sanov’s theorem, Schatten class
MSC. Primary 46B20, 60F10; Secondary 47B10, 52A23

###### Contents

  1. 1 Introduction
    1. 1.1 The large and moderate deviations perspective
  2. 2 An outrageously short introduction to large deviations theory
    1. 2.1 Motivation
    2. 2.2 Cramér type large deviations
    3. 2.3 Sanov’s theorem for the empirical measure
    4. 2.4 The Gärtner–Ellis theorem for dependent random variables
    5. 2.5 The abstract theory of large deviations
      1. 2.5.1 Three fundamental principles
  3. 3 Large deviations results and techniques in the world of $\ell_{p}^{n}$-spaces
    1. 3.1 Large deviations for random projections of $\ell_{p}^{n}$-balls
      1. 3.1.1 Annealed LDP – the case $p\in[2,\infty)$
      2. 3.1.2 Annealed LDP – the case $p\in[1,2)$
    2. 3.2 Large deviations for $\ell_{q}$-norms of random vectors in $\ell_{p}$-balls
    3. 3.3 Moderate deviations for $\ell_{q}$-norms of random vectors in $\ell_{p}$-balls
    4. 3.4 Large deviations, moderate deviations, and the KLS conjecture
  4. 4 Large deviations results and techniques in the world of Schatten classes
    1. 4.1 Self-adjoint and non self-adjoint Schatten $p$-classes
    2. 4.2 A non-commutative Schechtman–Zinn probabilistic representation
    3. 4.3 Sanov-type LDPs for random matrices in Schatten unit balls
      1. 4.3.1 LDPs in the self-adjoint case
      2. 4.3.2 LDPs in the non self-adjoint case
      3. 4.3.3 Strategy of the proof
  5. 5 Large deviations results and techniques in the world of Orlicz spaces
    1. 5.1 Orlicz spaces
    2. 5.2 The maximum entropy principle & Gibbs measures
    3. 5.3 Volumetric properties of Orlicz balls
    4. 5.4 Sanov-type large deviations
  6. 6 Large deviations results under an asymptotic thin-shell condition
    1. 6.1 The crucial regimes and asymptotic thin-shell-type assumptions
    2. 6.2 Main result in the constant regime
    3. 6.3 Main results in the sub-linear regime
    4. 6.4 Main results in the linear regime
    5. 6.5 Main examples satisfying asymptotic thin-shell-type assumptions
      1. 6.5.1 The case of product measures
      2. 6.5.2 The case of $\ell_{p}^{n}$-balls
      3. 6.5.3 The case of Orlicz balls

##  1 Introduction

Already the early years in the local theory of Banach spaces and geometric
functional analysis have demonstrated a deep connection between the geometry
of high-dimensional normed spaces, and thus of high-dimensional convex bodies,
and probability theory. This fruitful interplay has already been nicely
displayed in the wonderful and influential monographs _Asymptotic theory of
finite-dimen-sional normed spaces_ by V. D. Milman and G. Schechtman [88],
_The volume of convex bodies and Banach space geometry_ by G. Pisier [93], and
_Probability in Banach Spaces_ by M. Ledoux and M. Talagrand [84], and has
recently taken a new direction as nicely presented in [114]; we do not even
make the attempt here to list the numerous influential and groundbreaking
papers in this direction. Even though high-dimensional structures and
randomness seem inevitably to lead to the existence of more or too many
possibilities, resulting in an increased diversity and complexity as the
dimension increases, the past decades have shown that this is often
compensated and even patterns are created and concentration phenomena evoked,
its source being convexity. Along the way, powerful methods have been
developed by a large community of gifted mathematicians, both on the geometric
and analytic, as well as on the probabilistic side. In fact, new fields at the
crossroads between functional analysis, discrete and convex geometry, and
probability theory have emerged, today known as asymptotic geometric analysis
(also asymptotic convex geometry) and high-dimensional probability theory. Due
to their common origin both fields overlap in many different ways, while still
leaning towards the mathematical disciplines the names already indicate. The
methodical diversity, drawing on ideas from analysis, geometry, and
probability, is a common characteristic for both. Today the methods have
become indispensable in numerous problems of (multivariate) statistics,
computer science, theoretical numerical analysis, or compressed sensing, and
quite often research activities are driven by questions that arose in the
applied sciences, for instance, the famous Kannan–Lovász–Simonovits conjecture
[69], which stems from an algorithmic question about the complexity of volume
computation of a high-dimensional convex body (we will encounter this
conjecture again later). We refer to the monographs [9, 10] and [22], which
provide an outstanding and extensive overview of modern asymptotic geometric
analysis until the year $2021$, the recent excellent introduction to high-
dimensional probability with applications in data science [114], and the
comprehensive textbook [36] about the mathematical aspects of compressed
sensing, all of which are complemented by their nearly exhaustive lists of
references. In particular, the monographs [9, 10, 22] treat the classical
central limit perspective, most prominently Klartag’s central limit theorem
for convex bodies [76], saying that given any isotropic convex body in high
dimension its typical random projections will be approximately standard
Gaussian (see [7] for the origin of the problem). This is complemented by
other central limit theorems such as [4], [58], [63], [91], or [106], just to
mention a few (please consult the respective lists of references for more in
this direction). In this manuscript, we shall be concerned with other types of
limit theorems going beyond the normal fluctuations covered by a central limit
theorem.

###  1.1 The large and moderate deviations perspective

The beautiful universality that emerges from the central limit perspective
unfortunately restricts the information that may be retrieved, for instance,
from lower-dimensional projections of a high-dimensional convex body. But
which aspects of, for instance, random projections can be used to distinguish
between different convex bodies?

In order to answer this question, let us recall that a sequence
$(X_{n})_{n\in\mathbb{N}}$ of random vectors in $\mathbb{R}^{d}$ satisfies a
large deviations principle (LDP) with lower-semicontinuous rate function
$\operatorname{\mathbb{I}}$ and speed $s_{n}\uparrow\infty$ if and only if for
all Borel sets $A\subset\mathbb{R}^{d}$

$\mathbb{P}[X_{n}\in A]\approx e^{-s_{n}\inf_{x\in
A}\operatorname{\mathbb{I}}(x)}.$

A moderate deviations principle (MDP) formally resembles an LDP but on a
different scale and, typically, with significant differences in the behavior.
More precisely, the scale for an MDP of a random quantity is between that of a
weak limit theorem (like a central limit theorem) and that of a law of large
numbers. It is instructive to consider the case of independent and identically
distributed (iid) $\mathbb{R}$-valued random variables $X_{i}$,
$i\in\mathbb{N}$, with finite exponential moments for which Cramér’s theorem
[28, Theorem I.4] guarantees that, for all $a>\mathbb{E}[X_{1}]$,

$\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}\Big{[}\frac{1}{n}\sum_{i=1}^{n}X_{n}\geq
a\Big{]}=-\operatorname{\mathbb{I}}(a),$

with $\operatorname{\mathbb{I}}$ being the Legendre transform of the log-
moment generating function of $X_{1}$. While this LDP occurs at scale $n$ of a
law of large numbers, as seen from the expression
$\frac{1}{n}\sum_{i=1}^{n}X_{i}$, an MDP occurs (under suitable conditions,
e.g., [31]) on a scale between $\sqrt{n}$ and $n$, more precisely,
$\frac{1}{b_{n}}\sum_{i=1}^{n}X_{i}$ with $b_{n}/n\to 0$ and
$b_{n}/\sqrt{n}\to\infty$.

In contrast to a central limit theorem, large and moderate deviations
principles are known to be typically non-universal and distribution-dependent,
more precisely, they are parametric in speed and/or rate function; MDPs are
usually less parametric in the rate (up to a variance dependence that may
appear). In this sense, the large and moderate deviations behavior of, for
instance, a random projection of a convex body, depends in a subtle way on the
geometry of the underlying body. In other words, LDPs and MDPs allow one to
distinguish high-dimensional probability measures, like the uniform
distributions on a high-dimensional convex body, via their lower-dimensional
projections. We shall describe below the development of this large and
moderate deviations approach in the asymptotic theory of geometric functional
analysis, but also the influence of large deviations (and in particular
statistical mechanics) ideas and techniques, before we move on with a short
introduction to large deviations theory and a more detailed presentation and
discussion of some of the results presented now.

The contributions we have seen and the progress that has been made in the past
seven years started with an inspiring work of N. Gantert, S. S. Kim, and K.
Ramanan [39] who studied the question about the large deviations behavior of
one-dimensional marginals of (scaled) $\ell_{p}^{d}$-balls defined as
${\mathbb{B}}_{p}^{d}:=\\{x\in\mathbb{R}^{d}\,:\,\|x\|_{p}\leq 1\\}$, $1\leq
p\leq\infty$, both in an annealed (i.e., marginal in a random direction) and
quenched (i.e., marginal in a fixed deterministic direction) setting. In the
annealed setting, their work reveals a stark difference in the behavior both
speed- and rate-wise, depending on whether $1\leq p\leq 2$ or $2<p\leq\infty$.
The authors also identify a variational formula that relates the annealed and
quenched LDPs.

This is where D. Alonso-Gutiérrez, J. Prochno, and C.Thäle started, looking at
higher-dimensional marginals, studying LDPs for the Euclidean norm of random
orthogonal $k_{d}$-dimensional projections of random vectors uniformly
distributed on $\ell_{p}^{d}$-balls; here $k_{d}\in\mathbb{N}$ may tend to
infinity as $d\to\infty$. The observations are two-fold. On the one hand, we
see again a significantly different large deviations behavior, depending on
whether $1\leq p\leq 2$ or $2<p\leq\infty$. On the other hand, the form of the
rate function that governs the LDP depends on the asymptotic behavior of the
random subspace dimensions $k_{d}$ relative to $d$. One should particularly
notice that for $1\leq p<2$, both the speed and the rate function differ from
those for the $1$-dimensional random projections in [39]. Notably, when $p=2$
the LDP does not feel the dimensions $k_{d}$ of the random subspaces as long
as they do not increase too fast with $d$ and we obtain the same LDP as in
[39]. The difference to the $1$-dimensional projections becomes visible only
in the truly high-dimensional regime in which $k_{d}$ is eventually
proportional to the dimension $d$ of the ambient space. When $2<p\leq\infty$
the LDP occurs at the same speed as in [39], but with a different rate
function.

The case of general $\ell_{q}$-norms of random vectors uniformly distributed
on ${\mathbb{B}}_{p}^{d}$, i.e., without involving any projections, was
studied by Z. Kabluchko, J. Prochno, and C. Thäle in [63]. The LDPs obtained
in [63] significantly differ depending on the position of $p$ relative to $q$.
It is shown that when $q\leq p$, the speed is $d$ and the rate function feels
the radial part of the uniform distribution on ${\mathbb{B}}_{p}^{d}$, while a
lack of exponential moments makes the analysis for $q>p$ more delicate,
resulting in a slower speed $d^{p/q}$ and a non-convex rate function. In
particular, the LDP in this case is not susceptible to the radial component.
In the follow-up paper [67], the same authors studied $\ell_{q}$-norms of
random vectors with distributions from a more general class related to
$\ell_{p}$-balls. In particular, they considered MDPs for those random
quantities in the case where $q<p$. A somehow related work, where the random
points are chosen uniformly at random in a centered regular simplex in high
dimensions, is [11]. There the authors obtained Berry–Esseen bounds in the
regime $1\leq q<\infty$, complemented those by a non-central limit theorem
together with moderate and large deviations in the case where $q=\infty$, and
compared those results to the previously mentioned ones for the crosspolytope
${\mathbb{B}}_{1}^{d}$. We also refer to the work [47] of J. Grote, Z.
Kabluchko, and C. Thäle for several types of limit theorems, including MDPs
and LDPs, for high-dimensional random simplices.

The results mentioned so far are often referred to as level–1 large
deviations. The level–2 large deviations or Sanov-type large deviations
concern LDPs for random empirical measures. In [74], S. S. Kim and K. Ramanan
studied conditional limit theorems for $\ell_{p}$-spheres in the flavor of the
classical and famous Poincaré–Maxwell–Borel Lemma (see [21] or [29]). Central
to their analysis is an LDP that the authors establish for the sequence of
empirical measures of the coordinates of a random vector uniformly distributed
on the sphere
${\mathbb{S}}_{p}^{d-1}:=\\{x\in\mathbb{R}^{d}\,:\,\|x\|_{p}=1\\}$ (both,
random vector and sphere are scaled with a factor $d^{1/p}$). Notably, the LDP
not only holds in the weak topology (on the space of probability measures),
but even in the stronger $q$-Wasserstein topology as long as $q<p$. A nice
combination with another element of large deviations theory, the Gibbs
conditioning principle, allows them to obtain the conditional limit theorem.

Let us briefly stop here and explain a central tool in all the work presented
so far, which will then also serve as motivation for the research discussed in
a moment. In the world of $\ell_{p}^{d}$-balls, there exists an inconspicuous
probabilistic representation of the uniform measure on ${\mathbb{B}}_{p}^{d}$,
relating it for any $1\leq p<\infty$ to the so-called $p$-Gaussian
distribution with Lebesgue density of the form

$\mathbb{R}\ni x\mapsto C_{p}e^{-|x|^{p}/p},$

where $C_{p}\in(0,\infty)$ is a normalization constant. More precisely, the
work of G. Schechtman and J. Zinn [107] (see also the independent work [96] by
S. T. Rachev and L. Rüschendorf) shows that if $X$ is uniformly distributed on
${\mathbb{B}}_{p}^{d}$, and $Z_{1},\dots,Z_{d}$ are independent
$p$-generalized Gaussian random variables, then

$X\stackrel{{\scriptstyle\textup{d}}}{{=}}U^{1/d}\frac{(Z_{1},\dots,Z_{d})}{\|(Z_{1},\dots,Z_{d})\|_{p}},$

where $U$ is uniformly distributed on $[0,1]$ and independent of
$Z_{1},\dots,Z_{d}$. In view of a probabilistic approach to the geometry of
$\ell_{p}^{d}$-balls, this representation is quite powerful, because it allows
to make a transition from a random vector with dependent coordinates (like
$X$) to one with independent coordinates (like $(Z_{1},\dots,Z_{d})$); this
makes many problems computationally feasible. Behind any of the large and
moderate deviations results for $\ell_{p}^{d}$-balls stands this
Schechtman–Zinn probabilistic representation, also in [39].

Motivated by the work in [74], Z. Kabluchko, J. Prochno, and C. Thäle
investigated Sanov-type LDPs in the non-commutative setting of Schatten
classes [66], where no such Schechtman–Zinn representation had been available.
We recall that the Schatten $p$-class is the space of $d\times d$ real
matrices where the norm of a matrix $A$ is given by the $\ell_{p}^{d}$-norm of
the vector of singular values of $A$. The authors established a non-
commutative version of the Schechtman–Zinn representation and proved an LDP
for the (properly scaled) empirical spectral measure of a matrix chosen
uniformly at random from a unit ball in the Schatten $p$-class or from the
boundary with respect to the cone measure (on the space of probability
measures equipped with the weak topology). As a consequence they also obtained
that the spectral measure converges weakly almost surely to a non-random
limiting measure given by the Ullman distribution. It turns out that in this
non-commutative setting, the analysis is considerably more delicate. This
paper was the first going beyond the setting of $\ell_{p}$-balls. Moreover,
the ideas and results developed in [66] and the related works [64, 65] (see
also [108]), which largely stem from large deviations theory and statistical
mechanics, where crucial in [26] to establish a generalized version of the
variance conjecture for the family of Schatten $p$-class unit balls of self-
adjoint matrices.

Another work dealing at least in parts with the non-commutative setting is
[72]. T. Kaufmann and C. Thäle derive a probabilistic representation for
classes of weighted $p$-radial distributions, which, in essence, are based on
mixtures of a weighted cone probability measure and a weighted uniform
distribution on ${\mathbb{B}}_{p}^{d}$. The authors then prove LDPs for the
empirical measure of the coordinates of vectors chosen at random from
${\mathbb{B}}_{p}^{d}$ with respect to distributions from the weighted measure
class. Those distributions are then extended to $p$-balls in classical matrix
spaces, both for self-adjoint and non-self-adjoint matrices, and their
eigenvalue and singular value distributions are identified for the respective
random matrices. As an application, LDPs for the empirical spectral measures
for the eigenvalues and the singular values are proved.

In [68], Z. Kabluchko, J. Prochno, and C. Thäle proved a strong law of large
numbers for $k$-dimensional random projections of the $d$-dimensional cube via
the Artstein-Vitale law of large numbers for random compact sets. This result,
which can also be derived from the celebrated Dvoretzky–Milman theorem, says
that with respect to the Hausdorff distance, a uniform random projection of
the $d$-dimensional cube onto $\mathbb{R}^{k}$ converges almost surely to a
centered $k$-dimensional Euclidean ball of radius $\sqrt{2/\pi}$, as
$d\to\infty$. For every point inside this, the authors determined the
asymptotic number of vertices and the volume of the part of the cube projected
‘close’ to this point. This result is a consequence of a more general large
deviations result for random projections of arbitrary product measures
obtained in this paper.

Related to the $d$-dimensional cube is also a work of S. Johnston, Z.
Kabluchko, and J. Prochno, who studied the large deviations behavior of random
one-dimensional projections of the uniform distribution on the cube
$[-1,1]^{d}$ [57]. While a combination of Klartag’s central limit theorem for
convex bodies with the Borel–Cantelli lemma implies the almost sure weak
convergence to a centered Gaussian distribution of variance $1/3$, and thus
shows that the typical behavior is universal, the authors establish an LDP
showing that the atypical behavior is not universal. In fact, the speed of the
LDP is $d$ and the rate function is explicit, thus giving direct access to the
dependence on the underlying distribution.

The previous results from [57] have been considerably generalized by Z.
Kabluchko and J. Prochno in [61], where LDPs for random matrices in the
orthogonal group and Stiefel manifold are proved. The good convex rate
functions are explicitly given in terms of certain log-determinants of trace-
class operators and are finite on the set of Hilbert–Schmidt operators $M$
satisfying $\|MM^{*}\|<1$. Those results are then applied to determine the
precise large deviation behavior of $k$-dimensional random projections of
high-dimensional product distributions using an appropriate interpretation in
terms of point processes, also providing a characterizing of the space of all
possible deviations. The case of uniform distributions on
${\mathbb{B}}_{p}^{d}$, $1\leq p\leq\infty$, is also considered and reduced to
appropriate product measures.

A work partly generalizing the one of Z. Kabluchko and J. Prochno [61] is the
recent paper [71] by T. Kaufmann, H. Sambale, and C. Thäle. Here the authors
extend the set of projected distributions for which an LDP is shown (going
from the uniform distribution to quite general $p$-radial distributions) and
observe that the large deviations behavior obtained in [61] is universal for a
large class of probability measures on ${\mathbb{B}}^{d}_{p}$. They also
describe geometrically motivated distributions on the balls
${\mathbb{B}}^{d}_{p}$ for which the LDP needs a suitable modification.

Another work dealing with large deviations induced by the Stiefel manifold is
[75]. In this paper, S. S. Kim and K. Ramanan prove LDPs for $k$-dimensional,
deterministic Stiefel projections of classes of random vectors in
$\mathbb{R}^{d}$ ($k<d$ fixed) including uniform distributions on suitably
normalized $\ell_{p}^{d}$-balls with $1<p\leq\infty$ and product measures with
sufficiently light tails; the results hold under some convergence assumption
on the empirical measures of the properly scaled Stiefel matrix. The results
are then nicely applied to obtain large deviations for random projections in
the quenched sense, where the Stiefel matrices are chosen at random with
respect to the Haar measure on the Stiefel manifold and quenched refers to the
fact that one conditions on almost sure realizations of the random Stiefel
matrix. The authors also obtain a variational formula for the rate function
governing the LDP for annealed projections. It is important to note that the
key difference compared to [61] is in the object of study, which in [75] is
the projection point itself, hence yielding an LDP on $\mathbb{R}^{k}$,
whereas in [61] it is the projected distribution on $\mathbb{R}^{k}$ (and thus
the main result yields an LDP on the space of probability measures on
$\mathbb{R}^{k}$.

Having its origin in theoretical computer science, the
Kannan–Lovász–Simonovits conjecture (see, e.g., [1]) is arguably the most
famous open problem in asymptotic geometric analysis and high-dimensional
probability theory today; the currently best known result is due to B. Klartag
[78]. As shown by D. Alonso-Gutiérrez, J. Prochno, and C. Thäle [5], there is
a connection between this conjecture and the study of large and moderate
deviations for isotropic log-concave random vectors, therefore providing an
alternative possibility to tackle the conjecture by presenting a potential
route to disproving it. In their work, the authors study the moderate
deviations for the Euclidean norm of random orthogonally projected random
vectors in ${\mathbb{B}}_{p}^{d}$, leading to a number of interesting
observations, namely that the $\ell_{1}^{d}$-ball is already critical for the
new approach, that for $p\geq 2$ the rate function in the MDP undergoes a
phase transition, depending on whether the scaling is below the square-root of
the subspace dimensions or comparable, and finally that for $1\leq p<2$ and
comparable subspace dimensions, the rate function again displays a phase
transition depending on its growth relative to $d^{p/2}$.

While the work on Sanov-type large deviations in the non-commutative setting
of Schatten $p$-classes [66] was the first in which an LDP was derived without
the classical Schechtman–Zinn probabilistic representation, it took a while to
understand how to go beyond the framework of the (commutative) sequence spaces
$\ell_{p}$. Quite recently, Z. Kabluchko and J. Prochno used a principle
deeply manifested in large deviations theory and statistical mechanics, the
so-called maximum entropy or Gibbs principle for non-interacting particles, to
establish a probabilistic approach to the geometry of Orlicz balls [60]. More
precisely, they extracted from this maximum entropy principle an asymptotic
version of a probabilistic representation for the uniform distribution on an
Orlicz ball. This allowed them, among other things, to compute the precise
asymptotic volume of Orlicz balls (based on sharp large deviations techniques)
and to determine the threshold behavior for the volume of intersections of
Orlicz balls, a result extending the $\ell_{p}$-ball counterpart of G.
Schechtman and M. Schmuckenschläger [106]. In particular, it became clear that
the role of the Schechtman–Zinn probabilistic representation is replaced, in
an asymptotic form, by appropriate Gibbs distributions whose potentials are
given by Orlicz functions. Let us note that, in essence, those ideas have been
used by F. Barthe and P. Wolff in [17] to show for a large family of Orlicz
balls that they verify the R. Kannan, L. Lovász, and M. Simonovits spectral
gap conjecture.

S. T. Rachev and L. Rüschendorf [96] and A. Naor and D. Romik [90] unified
results obtained in the 1980s by P. Diaconis and D. Freedman [29],
establishing a connection between $\ell_{p}^{d}$-balls and a $p$-generalized
Gaussian distribution. In [59], S. Johnston and J. Prochno studied similar
questions in a significantly broader setting, looking at low-dimensional
projections of random vectors uniformly distributed on sets of the form
${\mathbb{B}}_{\phi,t}^{d}:=\\{(s_{1},\ldots,s_{d})\in\mathbb{R}^{d}\,:\,\sum_{i=1}^{d}\phi(s_{i})\leq
td\\}$, where $\phi:\mathbb{R}\to[0,\infty]$ is a function satisfying some
fairly mild conditions; this framework includes the simplex as well as the
$\ell_{p}^{d}$-balls for $p>0$ and the more general case of Orlicz balls.
Using an approach completely different from [96] and [90], which is again
based on ideas from the theory of large deviations and statistical mechanics,
specifically those centered around Cramér’s theorem, the Gibbs conditioning
principle, and Gibbs measures, the authors showed that there is a critical
parameter $t_{\text{crit}}$ at which there is a phase transition in the
behavior of the low-dimensional projections. More precisely, for
$t>t_{\text{crit}}$ the coordinates of projected random vectors sampled from
${\mathbb{B}}_{\phi,t}^{d}$ behave like independent uniform random variables,
but for $t\leq t_{\text{crit}}$ however the Gibbs conditioning principle comes
into play, and here there is a parameter $\beta_{t}>0$ (the so-called inverse
temperature) such that the coordinates are approximately distributed according
to a density proportional to $e^{-\beta\phi(s)}$. The general ideas of this
paper have been developed in parallel with the ones put forward by Z.
Kabluchko and J. Prochno in [60].

In the work [2], D. Alonso-Gutiérrez and J. Prochno studied the asymptotic
thin-shell concentration for random vectors uniformly distributed on Orlicz
balls. Their approach was inspired by the ideas developed in [60] and combined
with moderate deviations techniques. Among other things, this allowed the
authors to determine the precise asymptotic value of the isotropic constant of
Orlicz balls and to provide both upper and lower bounds on the probability of
such a random vector being in a thin shell whose radius is a constant multiple
of the dimension $\sqrt{d}$. In fact, they manage to prove that in certain
cases their bounds are best possible and improve upon the currently best known
general bound when the deviation parameter goes down to zero as the dimension
$n$ of the ambient space increases.

The work [74] of S. S. Kim and K. Ramanan on Sanov-type large deviations and
conditional limit theorems for $\ell_{p}$-spheres has recently been lifted to
the setting of Orlicz spaces by L. Frühwirth and J. Prochno in [37]; the
underlying idea using appropriate Gibbs distributions again stems from [60].
In the paper, a Sanov-type large deviations principle for the sequence of
empirical measures of vectors chosen uniformly at random from an Orlicz ball
is obtained and then, from this level-2 large deviations result, in a
combination with Gibbs conditioning, entropy maximization, and the Orlicz
version of the Poincaré–Maxwell–Borel lemma due to S. Johnston and J. Prochno
[59], the authors deduce a conditional limit theorem for high-dimensional
Orlicz balls. In geometric parlance, the latter shows that if $M_{1}$ and
$M_{2}$ are Orlicz functions, then random points in the $M_{1}$-Orlicz ball,
conditioned on having a small $M_{2}$-Orlicz radius, look like an
appropriately scaled $M_{2}$-Orlicz ball. In fact, it is shown that the
limiting distribution in their Poincaré–Maxwell–Borel lemma, and thus the
geometric interpretation, undergoes a phase transition depending on the
magnitude of the $M_{2}$-Orlicz radius.

A few years ago there was actually a breakthrough on the commutative front. S.
S. Kim, Y.-T. Liao, and K. Ramanan established both level–1 and level–2 large
deviations principles for sequences of random vectors under asymptotic thin-
shell-type conditions [73]. In its most elementary form, this condition
requires the sequence of Euclidean norms of the random vectors, normalized by
a factor $d^{-1/2}$, to satisfy an LDP at speed $d$ and with rate function
having a unique minimum. Notably, the authors manage to establish this
asymptotic thin-shell condition in the case of random vectors uniformly
distributed on Orlicz balls whose Orlicz functions are of super-quadratic
growth. This is very nice for several reasons, but most prominently is maybe
the following: the authors identify a similarity to Klartag’s famous central
limit theorem which follows from the classical thin-shell estimate. The
mentioned work is quite impressive and the goal in the near future will be to
establish the asymptotic thin-shell condition for other classes of random
vectors, for instance, for those chosen uniformly at random from unit balls of
Lorentz sequence spaces (these spaces are, besides Orlicz sequence spaces, the
most natural sequence space generalizations of $\ell_{p}$-spaces and also
belong to the important class of $1$-symmetric Banach spaces); we refer to
[62] for a recent and first probabilistic approach in this setting.

Last but not least, let us mention that in recent years sharp large deviations
results in the flavor of R. R. Bahadur and R. R. Ranga Rao [12] or V. V.
Petrov [92] have been obtained for several random geometric quantities. While
LDPs identify the asymptotic exponential decay rate of probabilities in terms
of speed and rate function, sharp large deviation estimates also provide the
subexponential prefactor in front of the exponentially decaying term, which is
not visible on the logarithmic scale of an LDP. In that sense, sharp large
deviations provide more accurate quantitative estimates in finite dimensions,
which is more in the spirit of the local theory of Banach spaces. This line of
research was initiated by Y.-T. Liao and K. Ramanan [85] and picked up by T.
Kaufmann in [70] and L. Frühwirth and J. Prochno in [38]. Since the techniques
differ in various aspects from classical large deviations techniques, we shall
not discuss those results here and rather refer the interested reader to the
references above.

##  2 An outrageously short introduction to large deviations theory

Let us continue with a brief introduction to large deviations theory and some
of its fundamental concepts. The systematic study and unified formalization of
large deviations theory, which in essence concerns the asymptotic behavior of
remote tails of sequences of probability distributions, was developed by S.
Varadhan in 1966 [111]. In fact, he was later awarded the prestigious Abel
Prize “for his fundamental contributions to probability theory and in
particular for creating a unified theory of large deviations” in 2007.

We shall state results here in their most elementary form (sketch some of
their proofs) and provide references to more general ones where needed.
However, the goal is to provide the uninitiated reader with a rough
understanding of the underlying ideas and principles that are required later
on. For a thorough introduction to this fascinating topic, we refer the reader
to the survey of S. Varadhan [112] or the wonderful monographs by A. Dembo and
O. Zeitouni [27], R. S. Ellis [34], F. den Hollander [28], or by F. Rassoul-
Agha and T. Seppäläinen [98] as well as the survey article by H. Touchette
[110].

###  2.1 Motivation

Assume we are given iid random variables $X_{1},X_{2},\dots$, say for
simplicity $\mathbb{E}[X_{1}]=0$ and $\mathrm{Var}[X_{1}]=\sigma^{2}>0$, and
let $S_{n}:=\sum_{i=1}^{n}X_{i}$, $n\in\mathbb{N}$, denote the sequence of
partial sums. The law of large numbers tells us that, for any
$\varepsilon\in(0,\infty)$,

$\lim_{n\to\infty}\mathbb{P}\Big{[}\Big{|}\frac{S_{n}}{n}\Big{|}\geq\varepsilon\Big{|}\Big{]}=0.$

Chebyshev’s inequality yields a convergence rate of $1/n$, but can we, at
least under sufficient moment conditions, improve upon that bound?

An event of the form $\\{|S_{n}|\geq n\varepsilon\\}$ is referred to as a
large deviation or rare event, compared to the normal scale
$\\{|S_{n}|\geq\sqrt{n}\varepsilon\\}$ considered in a weak limit theorem. By
a direct computation one can check that, for instance, a sequence of
independent standard Gaussians $g_{1},g_{2},\dots$ satisfies, for any $a>0$,

$\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}\Big{[}\sum_{i=1}^{n}g_{i}\geq
an\Big{]}=-\frac{a^{2}}{2},$

and that a sequence of independent Rademacher random variables
$\xi_{1},\xi_{2},\dots$ satisfies

$\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}\Big{[}\sum_{i=1}^{n}\xi_{i}\geq
an\Big{]}=-I(a),$

where $I:\mathbb{R}\to[0,\infty]$ is defined as

$I(x):=\begin{cases}\frac{1+x}{2}\log(1+x)+\frac{1-x}{2}\log(1-x)&:\,x\in[-1,1]\\\
\infty&:\,|x|>1.\end{cases}$

If $X_{1},X_{2},\dots$ denotes either the sequence of Gaussians or the one of
Rademachers, then both examples indicate a similar behavior of the form

$\mathbb{P}\Big{[}\sum_{i=1}^{n}X_{i}\geq an\Big{]}\approx
e^{-n\operatorname{\mathbb{I}}(a)}$

for an appropriate function
$\operatorname{\mathbb{I}}:\mathbb{R}\to[0,\infty]$ and $a>0$. In particular,
the rate of convergence is exponential and thus way faster than the rate $1/n$
suggested by Chebyshev’s inequality.

This raises the question whether there is a more general principle behind
this? Let us assume that the random variables $X_{1},X_{2},\dots$ are iid,
centered, and have all exponential moments, i.e., for all $t\in\mathbb{R}$,
the moment generating function satisfies

$\lambda_{X}(t):=\mathbb{E}\Big{[}e^{tX_{1}}\Big{]}<\infty.$

It follows from Markov’s inequality, together with the iid assumption, that,
for any $a,t>0$,

$\mathbb{P}\Big{[}S_{n}\geq an\Big{]}=\mathbb{P}\Big{[}e^{tS_{n}}\geq
e^{tan}\Big{]}\leq
e^{-tan}\mathbb{E}\Big{[}e^{tS_{n}}\Big{]}=e^{-atn}\prod_{i=1}^{n}\lambda_{X}(t)=e^{-n[ta-\log\lambda_{X}(t)]},$

and a simple optimization in the parameter $t>0$ shows that, for any $a>0$,

$\mathbb{P}\Big{[}S_{n}\geq an\Big{]}\leq
e^{-n\sup_{t>0}[ta-\log\lambda_{X}(t)]}.$

In particular,

$\limsup_{n\to\infty}\frac{1}{n}\log\mathbb{P}\Big{[}S_{n}\geq
an\Big{]}\leq-n\sup_{t>0}[ta-\log\lambda_{X}(t)],$

and the right-hand side is essentially the Legendre transform of the log-
moment generating function of $X_{1}$.

###  2.2 Cramér type large deviations

It turns out that the so-called Chernoff bound with the Legendre transform
presented above is actually optimal and this is the assertion of a classical
theorem due to H. Cramér [25]. Let us state this result here in its simplest
form.

######  Theorem 2.1 (H. Cramér, 1938).

Let $X_{1},X_{2},\dots$ be iid random variables such that, for all
$t\in\mathbb{R}$,

$\Lambda_{X}(t):=\log\mathbb{E}\Big{[}e^{tX_{1}}\Big{]}<\infty.$

Then, for every $a>\mathbb{E}[X_{1}]$,

$\lim_{n\to\infty}\log\mathbb{P}[S_{n}\geq an]=-\Lambda_{X}^{*}(a),$

where $\Lambda_{X}^{*}(a):=\sup_{s\in\mathbb{R}}[as-\Lambda_{X}(s)]$ is the
Legendre transform of $\Lambda_{X}$.

###### Idea of Proof..

First of all, it is easy to see that it is enough to consider the case
$\mathbb{E}[X_{1}]<0$ and $a=0$, i.e., it is enough to prove that

$\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}\Big{[}\sum_{i=1}^{n}X_{i}\geq
0\Big{]}=-\Lambda^{*}_{Y}(0).$

As we have seen, the upper bound follows essentially from the exponential
Markov inequality. To argue that the supremum is actually attained on
$(0,\infty)$ one needs to take the distribution of mass of $X_{1}$ into
account, which means consider separately the following cases: (1)
$\mathbb{P}[X_{1}<0]$, (2) $\mathbb{P}[X_{1}\leq 0]$ and
$\mathbb{P}[X_{1}=0]>0$ as well as (3) $\mathbb{P}[X_{1}<0]>0$ and
$\mathbb{P}[X_{1}>0]>0$. Note that we used here the reduction to
$\mathbb{E}[X_{1}]<0$.

To explain the idea behind the proof the lower bound, let us assume that for a
set of nice random variables $(Y_{i})_{i\in\mathbb{N}}$ and
$a>\mathbb{E}[Y_{1}]$, $\sigma^{2}=\mathrm{Var}[X_{1}]>0$, we are interested
in $\mathbb{P}[\sum_{i=1}^{n}Y_{i}\geq an]$. If we standarize the partial sum
of random variables, then we obtain

$\mathbb{P}\Big{[}\sum_{i=1}^{n}Y_{i}\geq
an\Big{]}=\mathbb{P}\Big{[}\frac{S_{n}-n\mathbb{E}[X_{1}]}{\sigma\sqrt{n}}\geq\frac{\sqrt{n}(a-\mathbb{E}[X_{1}])}{\sigma}\Big{]}.$

Now, for large $n\in\mathbb{N}$, Gaussian approximation tells us that

$\mathbb{P}\Big{[}\sum_{i=1}^{n}Y_{i}\geq
an\Big{]}\approx\frac{1}{\sqrt{2\pi}}\int_{\frac{\sqrt{n}(a-\mathbb{E}[X_{1}])}{\sigma}}^{\infty}e^{-y^{2}/2}\,\textup{d}y.$

Because of the $\sqrt{n}$ in the lower bound of the integration domain, this
Gaussian approximation is not good if $a-\mathbb{E}[X_{1}]$ is not small. The
idea, which actually goes back to F. Esscher [35], is to perform an
exponential tilt of the original measure (today known as Esscher transform or
Cramér transform) that adds more weight to larger values, thereby shifting the
mean to the right, which eliminates the lower bound of the integration domain.
As it turns out, the price that one pays in the integrand is acceptable and
the central limit theorem can be properly applied to obtain the lower bound in
Cramér’s theorem.

More formally, in our set-up of $\mu:=\mathbb{P}^{X_{1}}$, we define the
exponentially tilted measure to be

$\widetilde{\mu}(\textup{d}x):=\frac{e^{t_{\text{min}}x}}{\rho}\mu(\textup{d}x)$

with $\rho:=e^{-\Lambda_{X}^{*}(0)}=\inf_{t\in\mathbb{R}}\lambda_{X}(t)$ and
$t_{\textrm{min}}>0$ such that $\lambda_{X}(t_{\textrm{min}})=\rho$ and
$\lambda_{X}^{\prime}(t_{\textrm{min}})=0$. Given this measure, we consider
independent random variables $\widetilde{X}_{1},\widetilde{X}_{2},\dots$ with
distribution $\widetilde{\mu}$. In particular, those random variables satisfy
$\mathbb{E}[\widetilde{X}_{1}]=0=a$ and thus deal with the problematic part
mentioned before. Moreover, a change of measure from $\mu$ to
$\widetilde{\mu}$ shows that

$\mathbb{P}[S_{n}\geq
0]=\rho^{n}\mathbb{E}\Bigg{[}e^{t_{\text{min}}\widetilde{S}_{n}}\mathbbm{1}_{\\{\widetilde{S}_{n}\geq
0\\}}\Bigg{]},$

where $\widetilde{S}_{n}:=\sum_{i=1}^{n}\widetilde{X}_{i}$. Thus,

$\frac{1}{n}\log\mathbb{P}[S_{n}\geq
0]=\underbrace{\log(\rho)}_{=-\Lambda_{X}^{*}(0)}+\frac{1}{n}\log\mathbb{E}\Bigg{[}e^{t_{\text{min}}\widetilde{S}_{n}}\mathbbm{1}_{\\{\widetilde{S}_{n}\geq
0\\}}\Bigg{]}$

and it is therefore enough to show that the second summand on the right-hand
side is non-negative. But this follows from the classical central limit
theorem, since

$\displaystyle\frac{1}{n}\log\mathbb{E}\Bigg{[}e^{t_{\text{min}}\widetilde{S}_{n}}\mathbbm{1}_{\\{\widetilde{S}_{n}\geq
0\\}}\Bigg{]}$
$\displaystyle\geq\frac{1}{n}\log\mathbb{E}\Bigg{[}e^{-t_{\text{min}}\widetilde{S}_{n}}\mathbbm{1}_{\\{0\leq\widetilde{S}_{n}\leq
1\\}}\Bigg{]}$
$\displaystyle\geq-\frac{t_{\text{min}}\sqrt{n}}{n}+\frac{1}{n}\log\mathbb{P}\Bigg{[}\frac{\widetilde{S}_{n}}{\sqrt{n}}\in[0,1]\Bigg{]}\,\,\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0,$

which then completes the proof. ∎

A natural question that arises is the one of what happens if we are not in the
luxurious situation where we are working with random variables having all
exponential moments. What happens, for instance, if we are faced with a
sequence of heavy tailed random variables (those naturally appear, e.g., when
studying large deviations related to $\ell_{p}$-balls)? As it turns out, we
can still prove a version of Cramér’s theorem, also known as _Cramér’s theorem
for random variables with stretched exponential tails_.

######  Theorem 2.2 (Cramér for stretched exponential tails).

Consider iid and integrable random variables $X_{1},X_{2},\dots$. Assume that
there exist constants $r\in(0,1)$, $c\in(0,\infty)$, and
$0<C_{1}<C_{2}<\infty$ such that, for all $t\geq 0$,

$C_{1}e^{-ct^{r}}\leq\mathbb{P}[X_{1}\geq t]\leq C_{2}e^{-ct^{r}}.$ (1)

Then, for any $a>\mathbb{E}[X_{1}]$, we have that

$\lim_{n\to\infty}\frac{1}{n^{r}}\log\mathbb{P}[S_{n}\geq
an]=-c\big{(}a-\mathbb{E}[X_{1}]\big{)}^{r}.$

The property of stretched exponential tails is expressed through (1). In
particular, one can easily check that the lower bound in (1) guarantees that
$\mathbb{E}[e^{tX_{1}}]=\infty$ for all $t>0$. Note that the speed is $n^{r}$,
which is much slower than the speed $n$ in Cramér’s theorem (see Theorem 2.1).
Moreover, the rate function on the right-hand side is obviously non-convex,
i.e., it does not arise as a Legendre transform. Since a proof is not
contained in the standard literature, we provide at least a sketch here.

###### Idea of Proof of Theorem 2.2..

Let $a>0$. Without loss of generality we may assume that
$\mathbb{E}[X_{1}]=0$. We shall start with the lower bound, where one uses the
philosophy that a sum of iid random variables with stretched exponential tails
becomes large most easily if one of the random variables does, while all
others have ‘normal’ size. Lower bound: Let $\varepsilon>0$. Then, for each
$n\in\mathbb{N}$, we obtain from

$X_{1}\geq
n(a+\varepsilon)\qquad\text{and}\qquad\sum_{i=2}^{n}X_{i}\geq-(n-1)\varepsilon$

that

$S_{n}=\sum_{i=1}^{n}X_{i}\geq
n(a+\varepsilon)-(n-1)\varepsilon=an+\varepsilon.$

Hence, for all $n\in\mathbb{N}$,

$\displaystyle\mathbb{P}[S_{n}\geq an]$ $\displaystyle=\mathbb{P}[X_{1}\geq
n(a+\varepsilon)]\cdot\mathbb{P}\Big{[}\frac{S_{n-1}}{n-1}\geq-\varepsilon\Big{]},$

where we used the iid property of the sequence $X_{1},X_{2},\dots$. As a
consequence of the weak law of large numbers,

$\mathbb{P}\Big{[}\frac{S_{n-1}}{n-1}\geq-\varepsilon\Big{]}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}1.$ (2)

Thus, the lower bound in (1) yields that, for any $n\in\mathbb{N}$,

$\displaystyle\frac{1}{n^{r}}\log\mathbb{P}[S_{n}\geq an]$
$\displaystyle\geq\frac{1}{n^{r}}\log
C_{1}-c(a+\varepsilon)^{r}+\frac{1}{n^{r}}\log\mathbb{P}\Big{[}\frac{S_{n-1}}{n-1}\geq-\varepsilon\Big{]}\,.$

This together with (2) shows that

$\liminf_{n\to\infty}\frac{1}{n^{r}}\log\mathbb{P}[S_{n}\geq
an]\geq-c(a+\varepsilon)^{r},$

which completes the proof of the lower bound for $\varepsilon\downarrow 0$.
Upper bound: The upper bound requires tedious computations, which we shall
shorten considerably, leaving it to the reader to fill in the details. First,
we note that by the very same philosophy already used in the lower bound, for
all $n\in\mathbb{N}$,

$\displaystyle\mathbb{P}[S_{n}\geq an]$
$\displaystyle=\mathbb{P}\Big{[}S_{n}\geq an\,,\,\max_{1\leq i\leq n}X_{i}\geq
an\Big{]}+\mathbb{P}\Big{[}S_{n}\geq an\,,\,\max_{1\leq i\leq
n}X_{i}<an\Big{]}$ $\displaystyle\leq\mathbb{P}\Big{[}\max_{1\leq i\leq
n}X_{i}\geq an\Big{]}+\mathbb{P}\Big{[}S_{n}\geq an\,,\,\max_{1\leq i\leq
n}X_{i}<an\Big{]}.$

We will show that the second summand does not dominate the first on the scale
$n^{r}$.

Because of the identical distribution of the random variables, for the first
summand we have

$\mathbb{P}\Big{[}\max_{1\leq i\leq n}X_{i}\geq
an\Big{]}\leq\sum_{i=1}^{n}\mathbb{P}[X_{i}\geq an]=n\mathbb{P}[X_{1}\geq
an].$

In combination with the upper bound in (1) this yields, for any
$n\in\mathbb{N}$,

$\displaystyle\frac{1}{n^{r}}\log\mathbb{P}\Big{[}\max_{1\leq i\leq
n}X_{i}\geq an\Big{]}$ $\displaystyle\leq\frac{1}{n^{r}}\log
n+\frac{1}{n^{r}}\log C_{2}-ca^{r}$

and so see that

$\limsup_{n\to\infty}\frac{1}{n^{r}}\log\mathbb{P}\Big{[}\max_{1\leq i\leq
n}X_{i}\geq an\Big{]}\leq-ca^{r}.$ (3)

Using the simple fact that, for all $N\in\mathbb{N}$ and every sequence
$a_{1}(n),\dots,a_{N}(n)\geq 0$, $n\in\mathbb{N}$,

$\limsup_{n\to\infty}\,\frac{1}{n^{r}}\,\log\sum_{i=1}^{N}a_{i}(n)=\max_{1\leq
i\leq N}\limsup_{n\to\infty}\frac{1}{n^{r}}\,\log a_{i}(n),$

we obtain for $N=2$ in combination with (3) that

$\displaystyle\limsup_{n\to\infty}\frac{1}{n^{r}}\log\mathbb{P}[S_{n}\geq an]$
$\displaystyle\leq\max\bigg{\\{}-ca^{r}\,,\,\limsup_{n\to\infty}\frac{1}{n^{r}}\log\mathbb{P}\Big{[}S_{n}\geq
an\,,\,\max_{1\leq i\leq n}X_{i}<an\Big{]}\bigg{\\}}\,.$

Hence, it is enough to prove that the second entry in the maximum does not
dominate the first. Using Markov’s inequality, we obtain, for every
$\alpha>0$,

$\displaystyle\mathbb{E}\bigg{[}\prod_{i=1}^{n}e^{\alpha
X_{i}}\mathbbm{1}_{\\{X_{i}<na\\}}\bigg{]}$ $\displaystyle\geq e^{\alpha
na}\,\mathbb{P}\Big{[}S_{n}\geq an\,,\,\max_{1\leq i\leq n}X_{i}<an\Big{]}.$

Equivalently, for every $\alpha>0$, we have

$\mathbb{P}\Big{[}S_{n}\geq an\,,\,\max_{1\leq i\leq n}X_{i}<an\Big{]}\leq
e^{-\alpha na}\,\mathbb{E}\bigg{[}\prod_{i=1}^{n}e^{\alpha
X_{i}}\mathbbm{1}_{\\{X_{i}<na\\}}\bigg{]}$

and so the iid assumption yields that

$\displaystyle\mathbb{P}\Big{[}S_{n}\geq an\,,\,\max_{1\leq i\leq
n}X_{i}<an\Big{]}$ $\displaystyle\leq e^{-\alpha
na}\bigg{(}\mathbb{E}\Big{[}e^{\alpha
X_{1}}\mathbbm{1}_{\\{X_{1}<\alpha^{-1}\\}}\Big{]}+\mathbb{E}\Big{[}e^{\alpha
X_{1}}\mathbbm{1}_{\\{\alpha^{-1}\leq X_{1}<an\\}}\Big{]}\bigg{)}^{n}.$

We will now show that for appropriate choice of $\alpha$, the factor
$e^{-\alpha na}$ has the desired rate $-ca^{r}$. _The first expectation:_
Using that for all $x<1$, $e^{x}\leq 1+x+x^{2}$, and that, for all
$x\in\mathbb{R}$, $1+x\leq e^{x}$, we obtain from a direct computation that
(since $\mathbb{E}[X_{1}]=0$ and $\sigma^{2}=\mathbb{E}[X_{1}^{2}]$)

$\displaystyle\mathbb{E}\Big{[}e^{\alpha
X_{1}}\mathbbm{1}_{\\{X_{1}<\alpha^{-1}\\}}\Big{]}$ $\displaystyle\leq
e^{\alpha^{2}\sigma^{2}}\,.$ (4)

_The second expectation:_ It follows from a tedious computation involving
integration by parts for Lebesgue-Stieltjes integrals that

$\mathbb{E}[e^{\alpha\xi}\mathbbm{1}_{\\{a\leq\xi\leq b\\}}]=e^{\alpha
a}\mathbb{P}[\xi\geq a]-e^{\alpha
b}\mathbb{P}[\xi>b]+\alpha\int_{a}^{b}e^{\alpha
x}\mathbb{P}[\xi>x]\,\textup{d}x$

holds for any random variable $\xi$ and all $a<b$. In our setting, together
with the upper bound in (1), we obtain

$\displaystyle\mathbb{E}\Big{[}e^{\alpha X_{1}}\mathbbm{1}_{\\{\alpha^{-1}\leq
X_{1}<an\\}}\Big{]}$ $\displaystyle\leq
C_{2}\alpha\int_{\alpha^{-1}}^{an}e^{\alpha
x-cx^{r}}\,\textup{d}x+C_{2}e^{1-c\alpha^{-r}},$

with absolute constant $C_{2}\in(0,\infty)$. Now we choose
$\alpha:=(c-\varepsilon)(an)^{r-1}$ with $\varepsilon\in(0,c)$. Then, for all
$x\in[\alpha^{-1},an]$, $\alpha x-cx^{r}\leq-\varepsilon x^{r}$ (where we used
$r\in(0,1)$). Thus, for sufficiently large $n\in\mathbb{N}$,

$\displaystyle\mathbb{E}\Big{[}e^{\alpha X_{1}}\mathbbm{1}_{\\{\alpha^{-1}\leq
X_{1}<an\\}}\Big{]}$ $\displaystyle\leq
C_{2}\alpha\int_{\alpha^{-1}}^{an}e^{-\varepsilon
x^{r}}\,\textup{d}x+e^{-Cn^{r(1-r)}},$

where $C:=C(a,c,C_{2},\varepsilon,r)\in(0,\infty)$. This means that, for
sufficiently large $n\in\mathbb{N}$,

$\displaystyle\mathbb{E}\Big{[}e^{\alpha X_{1}}\mathbbm{1}_{\\{\alpha^{-1}\leq
X_{1}<an\\}}\Big{]}$ $\displaystyle\leq
C_{2}\alpha\frac{1}{\varepsilon^{1/r}r}\Gamma(r^{-1};\varepsilon\alpha^{-r})+e^{-Cn^{r(1-r)}},$

where $\Gamma(\cdot;\cdot)$ is the incomplete gamma function. Using the
asymptotic $\Gamma(x;y)\sim y^{x-1}e^{-y}$ for $y\to\infty$, we obtain for
sufficiently large $n\in\mathbb{N}$ (note that $\alpha=\alpha(n)$) that

$\displaystyle\mathbb{E}\Big{[}e^{\alpha X_{1}}\mathbbm{1}_{\\{\alpha^{-1}\leq
X_{1}<an\\}}\Big{]}\leq e^{-\widetilde{C}n^{r(1-r)}},$ (5)

with $\widetilde{C}:=\widetilde{C}(a,c,C_{2},\varepsilon,r)\in(0,\infty)$.

_Completion of the proof:_ We will use the bounds (4) and (5). For the choice
$\alpha=(c-\varepsilon)(an)^{r-1}$ and sufficiently large $n\in\mathbb{N}$, we
obtain

$\displaystyle\mathbb{P}\Big{[}S_{n}\geq an\,,\,\max_{1\leq i\leq
n}X_{i}<an\Big{]}$ $\displaystyle\leq e^{-\alpha
na}\bigg{(}e^{\alpha^{2}\sigma^{2}}+e^{-\widetilde{C}n^{r(1-r)}}\bigg{)}^{n}$
$\displaystyle\leq
e^{-(c-\varepsilon)(an)^{r}}e^{C_{1}n^{2r-1}}\bigg{(}1+e^{-\widetilde{C}n^{r(1-r)}}\bigg{)}^{n},$

where $C_{1}:=C_{1}(\sigma,c,\varepsilon,a,r)\in(0,\infty)$. Since

$\bigg{(}1+e^{-\widetilde{C}n^{r(1-r)}}\bigg{)}^{n}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}1,$

we have that, for large $n\in\mathbb{N}$,

$e^{C_{1}n^{2r-1}}\bigg{(}1+e^{-\widetilde{C}n^{r(1-r)}}\bigg{)}^{n}\leq
e^{2C_{1}n^{2r-1}}.$

Noting that $2r-1<r$ and so $2C_{1}n^{2r-1}=o(n^{r})$ for $n\to\infty$, we see
that

$\displaystyle\limsup_{n\to\infty}\frac{1}{n^{r}}\log\mathbb{P}\Big{[}S_{n}\geq
an\,,\,\max_{1\leq i\leq n}X_{i}<an\Big{]}$
$\displaystyle\leq\limsup_{n\to\infty}\frac{1}{n^{r}}\log\Big{(}e^{-(c-\varepsilon)(an)^{r}}e^{o(n^{r})}\Big{)}=-(c-\varepsilon)a^{r}.$

Taking $\varepsilon\downarrow 0$ completes the proof. ∎

######  Remark 2.3.

We chose to state this theorem here, because later we shall use a more general
version of such a result (where the constant can dependent on the parameter
$t$), which is due to N. Gantert, K. Ramanan, and F. Rembart [40]. But the
philosophy behind it is more easily explained in the situation considered in
Theorem 2.2.

###  2.3 Sanov’s theorem for the empirical measure

Another fundamental theorem in large deviations theory, aside from Cramér’s
theorem, is Sanov’s theorem. This result concerns so-called level–$2$ large
deviations for the empirical measure of random variables and goes back to I.
N. Sanov [101]. We shall present here a more general and today standard form,
following the presentation in [98].

Let $E$ be a Polish space which will always be endowed with the Borel
$\sigma$-field. Consider $\lambda\in\mathscr{M}_{1}(E)$, where
$\mathscr{M}_{1}(E)$ is the set of probability measures on $E$. Consider iid
and $E$-valued random variables $X_{1},X_{2},\dots$ having distribution
$\lambda$ and define the corresponding empirical measure via

$\mu_{n}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}},\qquad n\in\mathbb{N}.$

This is a random probability measure on $E$, i.e., each $\mu_{n}$ is an random
element of $\mathscr{M}_{1}(E)$.

######  Remark 2.4.

Let us recall some well known facts. Since $E$ is a Polish space,
$\mathscr{M}_{1}(E)$ is a Polish space itself in the weak topology generated
by $\mathscr{C}_{b}(E)$, the space of bounded continuous functions on $E$
taking values in $\mathbb{R}$.

$\mathscr{M}(E)$ will denote the space of signed measures on $E$ of which
$\mathscr{M}_{1}(E)$ is a subspace. Then $\mathscr{M}(E)$ and
$\mathscr{C}_{b}(E)$ are in duality via

$\langle\nu,f\rangle=\int_{E}f\,\textup{d}\mu$

for $\nu\in\mathscr{M}(E)$, $f\in\mathscr{C}_{b}(E)$.

Convergence of probability measures in the weak topology
$\sigma(\mathscr{M}(E),\mathscr{C}_{b}(E))$ is the same as the weak
convergence $\nu_{n}\stackrel{{\scriptstyle w}}{{\longrightarrow}}\nu$, which
holds by definition if and only if
$\langle\nu_{n},f\rangle\rightarrow\langle\nu,f\rangle$ for all
$f\in\mathscr{C}_{b}(E)$ 111Recall that the weak topology on $\mathscr{M}(E)$,
$\sigma(\mathscr{M}(E),\mathscr{C}_{b}(E))$, is the minimal topology under
which all functions $\nu\mapsto\langle\nu,f,\rangle$, $f\in\mathscr{C}_{b}(E)$
are continuous..

Sanov’s theorem describes the large deviations behavior of the distributions

$\rho_{n}:=\mathbb{P}\big{[}\mu_{n}\in B\big{]},\qquad
B\subset\mathscr{M}_{1}(E)\,\,\text{Borel set},$

of $\mu_{n}$, $n\in\mathbb{N}$. Prior to proving an LDP it is good to develop
a sense for a law of large numbers, which we shall do now before stating
Sanov’s theorem. First, we note that by the classical strong law of large
numbers, for all $f\in\mathscr{C}_{b}(E)$,

$\displaystyle\mu_{n}(f)$
$\displaystyle:=\mathbb{E}_{\mu_{n}}[f]=\frac{1}{n}\sum_{i=1}^{n}f(X_{i})\xrightarrow[n\to\infty]{\mathbb{P}-a.s.}\mathbb{E}[f(X_{1})]=\mathbb{E}_{\lambda}[f].$
(6)

Let us recall that a sequence $(g_{i})_{i\in\mathbb{N}}$ is said to determine
weak convergence on $\mathscr{M}_{1}(E)$ if and only if
$\nu_{n}\stackrel{{\scriptstyle w}}{{\longrightarrow}}\nu$ is equivalent to
the condition that, for all $i\in\mathbb{N}$,
$\langle\nu_{n},g_{i}\rangle\rightarrow\langle\nu,g_{i}\rangle$. Now, since
our Polish space is by definition separable, there exists a countable,
convergence determining collection of $\mathscr{C}_{b}(E)$-functions. Let us
denote those functions by $(f_{i})_{i\in\mathbb{N}}$ with
$f_{i}\in\mathscr{C}_{b}(E)$ for all $i\in\mathbb{N}$. Because of (6), for
every $i\in\mathbb{N}$,

$\mu_{n}(f_{i})\xrightarrow[n\to\infty]{\mathbb{P}-a.s.}\mathbb{E}_{\lambda}[f_{i}].$

In particular, because $(f_{i})_{i\in\mathbb{N}}$ determines weak convergence
on $\mathscr{M}_{1}(E)$, we obtain the following strong law of large numbers
for $(\mu_{n})_{n\in\mathbb{N}}$:

$\mu_{n}\xrightarrow[n\to\infty]{\mathbb{P}-a.s.}\lambda.$

Now that we have developed an understanding of the typical asymptotic behavior
of the sequence of empirical measures corresponding to $X_{1},X_{2},\dots$, we
shall look at the asymptotic probabilities of rare events treated in Sanov’s
theorem.

######  Theorem 2.5 (Sanov’s theorem).

Let $E$ be a Polish space and, for each $n\in\mathbb{N}$, let $\rho_{n}$ be
the distribution of the empirical measure

$\mu_{n}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}},$

corresponding to the iid sequence $(X_{i})_{i\in\mathbb{N}}$ having
distribution $\lambda\in\mathscr{M}_{1}(E)$. Then
$(\rho_{n})_{n\in\mathbb{N}}$ satisfies an LDP on $\mathscr{M}_{1}(E)$ at
speed $n$ with good convex rate function
$H(\cdot|\lambda):\mathscr{M}_{1}(E)\to[0,\infty]$ being the relative entropy

$H(\nu|\lambda):=\begin{cases}\int_{E}f\log
f\,\textup{d}\lambda&:\nu\ll\lambda\,\text{ and
}\,f=\frac{\textup{d}\nu}{\textup{d}\lambda}\\\
\infty&:\text{otherwise}.\end{cases}$

In words, Sanov’s theorem tells us that the probability that the empirical
measures resemble some distribution $\nu$ other than the underlying
distribution $\lambda$ behaves like $e^{-nH(\nu|\lambda)}$, i.e., the rate is
determined by the entropy of $\nu$ relative to $\lambda$. We shall not provide
a proof of Sanov’s theorem here, but rather refer the interested reader to the
nice exposition of the proof in [98].

###  2.4 The Gärtner–Ellis theorem for dependent random variables

Before presenting the general theory of large deviations, let us describe a
classical generalization of Cramér’s theorem due to J. Gärtner [42] and R. S.
Ellis [33], which shows that we may allow random sequences with a kind of
moderate dependence, and still have LDP behavior. More precisely, let us
consider random vectors $(Y_{n})_{n\in\mathbb{N}}$ taking values in
$\mathbb{R}^{d}$ and having moment generating functions

$\lambda_{n}(t):=\mathbb{E}\Big{[}e^{\langle t,Y_{n}\rangle}\Big{]},\quad
t\in\mathbb{R}^{d},\,n\in\mathbb{N};$

here $\langle\cdot,\cdot\rangle$ denotes the standard inner product on
$\mathbb{R}^{d}$. We shall assume throughout this subsection that the
following two conditions are satisfied: (GE1)
$\lim\limits_{n\to\infty}\frac{1}{n}\log\big{(}\lambda_{n}(nt)\big{)}=\Lambda(t)\in[-\infty,\infty]$
exists, (GE2) $0\in\mathcal{D}_{\Lambda}^{\circ}$, with
$\mathcal{D}_{\Lambda}:=\\{t\in\mathbb{R}^{d}\,:\,\Lambda(t)<\infty\\}$. Under
the previous assumptions, it is rather easy to show that $\Lambda$ is convex
and satisfies $\Lambda>-\infty$. Moreover, the Legendre transform
$\Lambda^{*}(\cdot):=\sup_{t\in\mathbb{R}^{d}}[\langle\cdot,t\rangle-\Lambda(t)]$
is a convex good rate function (i.e., not everywhere equal to $\infty$ with
compact sublevel sets).

Let us recall that a point $x\in\mathbb{R}^{d}$ is called an exposed point for
$\Lambda^{*}$ if and only if there exists a point $t\in\mathbb{R}^{d}$ such
that, for all $\mathbb{R}^{d}\ni y\neq x$,

$\Lambda^{*}(y)-\Lambda^{*}(x)>\langle y-x,t\rangle.$

This $t$ is the normal vector to an exposing hyperplane for the point
$x\in\mathbb{R}^{d}$. Note that, since $\Lambda^{*}$ is convex, for each
$x\in\mathcal{D}_{\Lambda^{*}}$ there exists $t\in\mathbb{R}^{d}$ such that
$\Lambda^{*}(y)-\Lambda^{*}(x)\geq\langle t,y-x\rangle$ for all
$y\in\mathbb{R}^{d}$, i.e., a supporting hyperplane at the point
$(x,\Lambda^{*}(x))$ of the graph of $\Lambda^{*}$. Thus, at an exposed point
there exists a supporting hyperplane sharing no other point with the graph.

######  Theorem 2.6 (Gärtner–Ellis theorem).

Let $(Y_{n})_{n\in\mathbb{N}}$ be a sequence of random vectors in
$\mathbb{R}^{d}$ such that (GE1) and (GE2) are satisfied, and let
$\mu_{n}:=\mathbb{P}^{Y_{n}}$, $n\in\mathbb{N}$. Then, for all closed sets
$C\subset\mathbb{R}^{d}$,

$\limsup_{n\to\infty}\frac{1}{n}\log\big{(}\mu_{n}(C)\big{)}\leq-\inf_{x\in
C}\Lambda^{*}(x),$

and, for all open sets $O\subset\mathbb{R}^{d}$,

$\liminf_{n\to\infty}\frac{1}{n}\log\big{(}\mu_{n}(O)\big{)}\geq-\inf_{x\in
O\cap E}\Lambda^{*}(x),$

where $E$ is the collection of exposed points of $\Lambda^{*}$ whose exposing
hyperplane belongs to $\mathcal{D}_{\Lambda}^{\circ}$.

Suppose that, in addition, $\Lambda$ is lower semi-continuous on
$\mathbb{R}^{d}$, differentiable on $\mathcal{D}_{\Lambda}^{\circ}$, and that
$\mathcal{D}_{\Lambda}=\mathbb{R}^{d}$ or $\Lambda$ is steep on the boundary
$\partial\mathcal{D}_{\Lambda}$ of $\mathcal{D}_{\Lambda}$ (meaning that
$\lim\limits_{t\rightarrow\partial D_{\Lambda}\colon t\in
D_{\Lambda}}\lVert\nabla\Lambda(t)\rVert_{2}=\infty$). Then we may replace
$O\cap E$ by $O$. In large deviations parlance, the sequence
$(\mu_{n})_{n\in\mathbb{N}}$ satisfies an LDP on $\mathbb{R}^{d}$ with speed
$n$ and rate function $\Lambda^{*}$.

The steepness assumption may be removed in the iid setting [27, Corollary
6.1.6], but is generally required for dependent random sequences. Moreover,
one can generalize the result to topological vector spaces and we refer to
[27, Section 4.5]. For a more detailed discussion, we refer the interested
reader to [27] and [28]; it is also discussed there how the Gärtner–Ellis
theorem implies both Cramér’s and Sanov’s theorem.

###  2.5 The abstract theory of large deviations

As already mentioned, a general theory of large deviations, going far beyond
the ideas behind Cramér’s theorem, was developed by S. Varadhan in [111]. So
let us explore on what principles this unified theory is build upon.

######  Definition 2.7 (Rate function and LDP).

Let $\mathscr{X}$ be a topological Hausdorff space and
$\mathscr{B}_{\mathscr{X}}$ the Borel $\sigma$-field on $\mathscr{X}$. A rate
function $\operatorname{\mathbb{I}}:\mathscr{X}\to[0,\infty]$ is a lower-
semicontinuous mapping with $\operatorname{\mathbb{I}}\not\equiv\infty$, which
is said to be a good rate function if the level sets
$\\{x\in\mathscr{X}\,:\,\operatorname{\mathbb{I}}(x)\leq a\\}$,
$a\in[0,\infty)$ are not only closed, but compact. The effective domain of the
rate function is given by

$\mathscr{D}_{\operatorname{\mathbb{I}}}=\big{\\{}x\in\mathscr{X}\,:\,\operatorname{\mathbb{I}}(x)<\infty\big{\\}}.$

Let $(\nu_{n})_{n\in\mathbb{N}}$ be a sequence of probability measures on
$\mathscr{B}_{\mathscr{X}}$. Further, let $(s_{n})_{n\in\mathbb{N}}$ be a
sequence of positive reals with $s_{n}\uparrow\infty$ and
$\operatorname{\mathbb{I}}:\mathscr{X}\to[0,\infty]$ be a lower semi-
continuous function. We say that $(\nu_{n})_{n\in\mathbb{N}}$ satisfies a
(full) large deviation principle (LDP) at speed $s_{n}$ and with rate function
$\operatorname{\mathbb{I}}$ if and only if

$\begin{split}-\inf_{x\in
A^{\circ}}\operatorname{\mathbb{I}}(x)\leq\liminf_{n\to\infty}{1\over
s_{n}}\log\nu_{n}(A)\leq\limsup_{n\to\infty}{1\over
s_{n}}\log\nu_{n}(A)\leq-\inf_{x\in\overline{A}}\operatorname{\mathbb{I}}(x)\end{split}$
(7)

for all Borel sets $A\subset\mathscr{X}$, where $A^{\circ}$ and $\overline{A}$
denote the interior and closure of $A$, respectively. We say that
$(\nu_{n})_{n\in\mathbb{N}}$ satisfies a weak LDP with speed $s_{n}$ and rate
function $\operatorname{\mathbb{I}}$ if the rightmost upper bound in (7) is
valid only for compact sets $A\subset\mathscr{X}$.

######  Remark 2.8.

The notion of an LDP for a sequence of random variables is obtained by
considering the LDP for the corresponding sequence of distributions.

######  Remark 2.9.

The estimates in (7), describing the LDP, can also be reformulated in terms of
an upper and lower large deviations bound holding for closed and open sets
respectively, which is very much in the spirit of the bounds in the
Gärtner–Ellis theorem (see Theorem 2.6 above).

######  Remark 2.10.

Let us remark that the rate function controlling the LDP is always unique. A
consequence of the rate function being good is that it attains its infimum on
closed sets. Moreover, since $\nu_{n}(\mathscr{X})=1$ for any
$n\in\mathbb{N}$, for the upper bound in (7) to hold it is necessary that
$\inf_{x\in\mathscr{X}}\operatorname{\mathbb{I}}(x)=0$. So when
$\operatorname{\mathbb{I}}$ is a good rate function, then there must exist an
element $x\in\mathscr{X}$ such that $\operatorname{\mathbb{I}}(x)=0$.

Note that on metric spaces $M$, one may check the lower-semicontinuity on
sequences, i.e., $\operatorname{\mathbb{I}}$ is lower-semicontinuous if and
only if, for all $x\in M$,

$\liminf_{x_{n}\to
x}\,\operatorname{\mathbb{I}}(x_{n})\geq\operatorname{\mathbb{I}}(x).$

######  Remark 2.11.

In view of the Portmanteau theorem, this definition reminds us of the weak
convergence of measures. However, to the uninitiated reader this definition
only mildly resembles what one might expect after having seen Cramér’s
theorem. Naively one is tempted to think of a definition like

$\lim_{n\to\infty}\frac{1}{s_{n}}\log\nu_{n}(A)=-\inf_{x\in
A}\operatorname{\mathbb{I}}(x).$

As it turns out, this is far to restrictive to allow us to build a rich and
powerful theory. Indeed, note that if $(\nu_{n})_{n\in\mathbb{N}}$ is a
sequence of non-atomic measures, then $\nu_{n}(\\{x\\})=0$ for any
$x\in\mathscr{X}$. But this means that
$\operatorname{\mathbb{I}}\equiv\infty$, which is not allowed in the
definition of a rate function. So we really need some topological restrictions
in the definition of an LDP.

As we shall see later, when discussing the work on large deviation principles
in the asymptotic theory of geometric functional analysis, one way of proving
the large deviation estimates in (7) is to first prove the upper bound for
compact sets or, in other words, to first prove a weak LDP. Heuristically,
this can work if most of the mass in $\mathscr{X}$ is concentrated (on an
exponential scale) on compact sets.

######  Definition 2.12 (Exponential tightness).

Let $\mathscr{X}$ be a Hausdorff topological space with Borel $\sigma$-field
$\mathscr{B}_{\mathscr{X}}$ and assume that $(s_{n})_{n\in\mathbb{N}}$is a
sequence of positive reals with $s_{n}\uparrow\infty$. We say that a sequence
$(\nu_{n})_{n\in\mathbb{N}}$ of probability measures on
$\mathscr{B}_{\mathscr{X}}$ is exponentially tight (at scale $s_{n}$) if and
only if for every $C\in(0,\infty)$ there exists a compact set
$K_{C}\subset\mathscr{X}$ such that

$\limsup_{n\to\infty}\frac{1}{s_{n}}\log\nu_{n}(K_{C}^{c})<C.$

######  Remark 2.13.

Let us note that exponential tightness of a sequence of measures is not
necessary for the sequence to satisfy an LDP with a good rate function. In
nice situations however, if we are working with locally compact Hausdorff
topological spaces $\mathscr{X}$ or with Polish spaces (i.e., a separable
completely metrizable topological space), then goodness of the rate function
implies exponential tightness (see, e.g., [27]).

As it turns out, the notion of exponential tightness is the key to lifting a
weak LDP to a full one (see, e.g., [27, Lemma 1.2.18]). At the same time, this
guarantees us a good rate function in the LDP.

######  Lemma 2.14.

Let $\mathscr{X}$ be a Hausdorff topological space with Borel $\sigma$-field
$\mathscr{B}_{\mathscr{X}}$ and $(\nu_{n})_{n\in\mathbb{N}}$ a sequence of
probability measures on $\mathscr{B}_{\mathscr{X}}$. Suppose that
$(\nu_{n})_{n\in\mathbb{N}}$ satisfies a weak LDP at speed $s_{n}$ and with
rate function $\operatorname{\mathbb{I}}$. If $(\nu_{n})_{n\in\mathbb{N}}$ is
exponentially tight at scale $s_{n}$, then it satisfies a full LDP at speed
$s_{n}$ and with rate function $\operatorname{\mathbb{I}}$. In particular, in
this case the rate function $\operatorname{\mathbb{I}}$ is a good rate
function.

####  2.5.1 Three fundamental principles

There are several useful tools in the theory of large deviations that allow
one to obtain an LDP on one space by a transformation of a known LDP on some
other. Such transformations seem rather innocent at first, but are not to be
underestimated in their power. Note that we will refrain from providing any
examples here, because we shall see all principles in action later when
presenting some of the works on large deviations elaborated upon in the
introduction.

The first stability result concerns the preservation of LDPs under continuous
mappings and is commonly known as the contraction principle. Since the proof
is quite short, we will present it here.

######  Lemma 2.15 (Contraction principle).

Let $\mathscr{X}$ and $\mathscr{Y}$ be Hausdorff topological spaces and let
$T:\mathscr{X}\to\mathscr{Y}$ be a continuous mapping. Denote their respective
Borel $\sigma$-fields by $\mathscr{B}_{\mathscr{X}}$ and
$\mathscr{B}_{\mathscr{Y}}$. If $(\nu_{n})_{n\in\mathbb{N}}$ is a sequence of
probability measures on $\mathscr{B}_{\mathscr{X}}$ satisfying an LDP at speed
$s_{n}$ with a good rate function
$\operatorname{\mathbb{I}}:\mathscr{X}\to[0,\infty]$, then the sequence of
pushforwards

$\mu_{n}:=\nu_{n}\circ T^{-1},\qquad n\in\mathbb{N}$

satisfies an LDP on the space $\mathscr{Y}$ at the same speed $s_{n}$ and with
good rate function $\mathbb{J}:\mathscr{Y}\to[0,\infty]$

$\mathbb{J}(y):=\inf_{x\in\mathscr{X},T(x)=y}\operatorname{\mathbb{I}}(x),$

where we agree that the infimum over the empty set is equal to $\infty$.

###### Proof.

Let us first argue that the function $\mathbb{J}$ is indeed a good rate
function. We start by showing that $\mathbb{J}\not\equiv\infty$. Since
$\operatorname{\mathbb{I}}$ is a (good) rate function,
$\mathscr{D}_{\operatorname{\mathbb{I}}}=\big{\\{}x\in\mathscr{X}\,:\,\operatorname{\mathbb{I}}(x)<\infty\big{\\}}\neq\emptyset.$
And so it follows from the definition of $\mathbb{J}$ that
$\mathscr{D}_{\mathbb{J}}=\\{y\in\mathscr{Y}\,:\,\mathbb{J}(y)<\infty\\}\neq\emptyset,$
hence $\mathbb{J}\not\equiv\infty$. We now show that the function $\mathbb{J}$
is good. Clearly, $\mathbb{J}$ is non-negative and, in view of Remark 2.10 and
the continuity of $T$, attains its infimum for every $y\in T(\mathscr{X})$ at
some point in $\mathscr{X}$. We shall use this to show that the level sets are
compact. Let $a\in[0,\infty)$. Since $\operatorname{\mathbb{I}}$ is good,
$\\{x\in\mathscr{X}\,:\,\operatorname{\mathbb{I}}(x)\leq a\\}$ is a compact
set in $\mathscr{X}$. Therefore, since $T$ is continuous, $T\big{(}\\{x\in
E\,:\,\operatorname{\mathbb{I}}(x)\leq a\\}\big{)}$ is a compact set in
$\mathscr{Y}$. But since

$\displaystyle
T\big{(}\big{\\{}x\in\mathscr{X}\,:\,\operatorname{\mathbb{I}}(x)\leq
a\big{\\}}\big{)}$
$\displaystyle=\big{\\{}T(x)\,:\,x\in\mathscr{X},\,\operatorname{\mathbb{I}}(x)\leq
a\big{\\}}=\big{\\{}y\in\mathscr{y}\,:\,\exists
x\in\mathscr{X}:T(x)=y\,\,\land\,\,\operatorname{\mathbb{I}}(x)\leq
a\big{\\}}$ $\displaystyle=\big{\\{}y\in\mathscr{Y}\,:\,\inf_{x\in
T^{-1}(y)}\operatorname{\mathbb{I}}(x)\leq
a\big{\\}}=\big{\\{}y\in\mathscr{Y}\,:\,\mathbb{J}(y)\leq a\big{\\}},$

which shows the compactness of level sets. We used that $\mathbb{J}$ attains
its minimum on closed sets in the equality from line one to line two. ∎

The next principle frequently used, often in combination with the contraction
principle, is the following one describing the large deviations behavior of
pairs of independent random objects. We shall restrict ourselves to the case
of Polish spaces, enough for the purpose of this paper. However, the result
can be formulated and proved for regular topological spaces (see, e.g., [27,
Exercise 4.2.7]), in which case one needs to additionally assume the
exponential tightness of measures; by Remark 2.13 this follows from the
goodness of rate functions in Polish spaces.

######  Lemma 2.16.

Let $E$ be a Polish space. Assume that $(\nu_{n})_{n\in\mathbb{N}}$ and
$(\mu_{n})_{n\in\mathbb{N}}$ are sequences of probability measures on $E$.
Suppose that for all $n\in\mathbb{N}$, $(X_{n},Y_{n})$ is distributed
according to the product measure $\nu_{n}\otimes\mu_{n}$ on
$\mathscr{B}_{E}\otimes\mathscr{B}_{E}$ (i.e., $X_{n}$ and $Y_{n}$ are
independent). Assume that $(\nu_{n})_{n\in\mathbb{N}}$ satisfies an LDP at
speed $s_{n}$ with good rate function $\operatorname{\mathbb{I}}_{X}$ and
$(\mu_{n})_{n\in\mathbb{N}}$ an LDP at speed $s_{n}$ with good rate function
$\operatorname{\mathbb{I}}_{Y}$. Then,
(a) $(X_{n},Y_{n})$, $n\in\mathbb{N}$, satisfies an LDP on $E\times E$ at
speed $s_{n}$ with good rate function $\mathbb{J}:E\times E\to[0,\infty]$,

$\mathbb{J}(x,y):=\operatorname{\mathbb{I}}_{X}(x)+\operatorname{\mathbb{I}}_{Y}(y).$

(b) If $T:E\times E\to F$ is a continuous mapping into another Polish space
$F$, then the sequence $Z_{n}:=T(X_{n},Y_{n})$, $n\in\mathbb{N}$, satisfies an
LDP on $F$ at speed $s_{n}$ with good rate function
$\operatorname{\mathbb{I}}_{Z}:F\to[0,\infty]$,

$\operatorname{\mathbb{I}}_{Z}(z)=\inf_{(x,y)\in E\times
E\atop{T(x,y)=z}}\operatorname{\mathbb{I}}_{X}(x)+\operatorname{\mathbb{I}}_{Y}(y).$

The last principle we shall present here concerns the possibility of
transferring an LDP from one sequence of random variables to another as long
as those sequences are close on an exponential scale. Let $M$ be a metric
space with distance d, equipped with its Borel $\sigma$-field
$\mathscr{B}_{M}$. We say that two sequences of $M$-valued random variables
$(X_{n})_{n\in\mathbb{N}}$ and $(Y_{n})_{n\in\mathbb{N}}$ are exponentially
equivalent at speed $s_{n}$ if and only if, for all $\delta\in(0,\infty)$,

$\limsup_{n\to\infty}\frac{1}{s_{n}}\log\mathbb{P}\big{[}\textup{d}(X_{n},Y_{n})>\delta\big{]}=-\infty.$

The next result shows that, as far as the LDP is concerned, exponentially
equivalent random variables cannot be distinguished (see, e.g., [27, Theorem
4.2.13]).

######  Lemma 2.17.

Let $M$ be a metric space equipped with its Borel $\sigma$-field
$\mathscr{B}_{M}$ and assume that $(X_{n})_{n\in\mathbb{N}}$,
$(Y_{n})_{n\in\mathbb{N}}$ are sequences of $M$-valued random variables. If
$(X_{n})_{n\in\mathbb{N}}$ satisfies an LDP at speed $s_{n}$ with good rate
function $\operatorname{\mathbb{I}}$ and $(X_{n})_{n\in\mathbb{N}}$ and
$(Y_{n})_{n\in\mathbb{N}}$ are exponentially equivalent, then
$(Y_{n})_{n\in\mathbb{N}}$ satisfies an LDP at the same speed $s_{n}$ with the
same good rate function $\operatorname{\mathbb{I}}$.

##  3 Large deviations results and techniques in the world of
$\ell_{p}^{n}$-spaces

In this section we shall present some of the large deviations results that
have been obtained within the framework of $\ell_{p}^{n}$-balls, of course
starting with the inspiring work of N. Gantert, S. S. Kim, and K. Ramanan [39]
on random one-dimensional projections of $\ell_{p}^{n}$-balls. As we have
already mentioned in the introduction, in this setting, computations are
facilitated by a probabilistic representation of the uniform distribution on
an $\ell_{p}^{n}$-ball. More precisely, G. Schechtman and J. Zinn proved in
[107] that a random vector $X$ uniformly distributed on
${\mathbb{B}}_{p}^{n}$, $1\leq p<\infty$, has a representation

$\displaystyle
X\stackrel{{\scriptstyle\textup{d}}}{{=}}U^{1/n}\frac{(Z_{1},\dots,Z_{n})}{\|(Z_{1},\dots,Z_{n})\|_{p}},$
(8)

where $U$ is uniformly distributed on $[0,1]$ and independent of
$Z_{1},\dots,Z_{n}$, which are iid $p$-generalized Gaussians, i.e., they have
Lebesgue density

$f_{p}:\mathbb{R}\to[0,\infty),\qquad
x\mapsto\frac{1}{2p^{1/p}\Gamma(1+1/p)}e^{-|x|^{p}/p}.$ (9)

This representation is powerful as it allows one to work with a random vector
having independent coordinates; in contrast to the coordinates of $X$, which
are obviously dependent. We refer the reader to the survey [95] for a
discussion of the probabilistic approach to the geometry of
$\ell_{p}^{n}$-balls.

###  3.1 Large deviations for random projections of $\ell_{p}^{n}$-balls

In [39], N. Gantert, S. S. Kim, and K. Ramanan started analyzing
distributional properties of the projections of high-dimensional random
vectors in $\ell_{p}^{n}$-balls on the scale of large deviations. On the
normal deviations scale, a central limit theorem for projections of convex
bodies, i.e., for $X^{(n)}$ sampled from a log-concave measure (for instance
the uniform measure on a convex body) that is also isotropic, had already been
obtained by B. Klartag [76], showing that for sufficiently large
$n\in\mathbb{N}$ and most $\theta^{(n)}\in{\mathbb{S}}_{2}^{n-1}$, the
projection of $X^{(n)}$ onto $\theta^{(n)}$ satisfies $\langle
X^{(n)},\theta^{(n)}\rangle\approx\mathscr{N}(0,1)$ in some suitable sense.

We shall look at the results obtained in the annealed case, where a quite
different large deviations behavior is revealed depending on whether $p<2$ or
$p\geq 2$. In particular, this shows an interesting sensitivity with respect
to the underlying distribution of random vectors, which can be attributed to
the stark difference in the geometry of $\ell_{p}^{n}$-balls in those two
regimes. More precisely, the authors showed that if for each $n\in\mathbb{N}$,
$\theta^{(n)}\in{\mathbb{S}}^{n-1}$ is a uniform random direction and
$X^{(n,p)}$ is an independent random point uniformly distributed in the
$\ell_{p}^{n}$-ball for some fixed $p\in[1,\infty]$, then the sequence of
rescaled random variables (the normalized scalar random projections)

$W^{(n,p)}:=n^{{1\over p}-{1\over 2}}\langle
X^{(n,p)},\theta^{(n)}\rangle=\frac{1}{n}\sum_{i=1}^{n}n^{1\over
p}X_{i}^{(n,p)}n^{1\over 2}\theta^{(n)}_{i},\quad n\in\mathbb{N},$

satisfies an LDP with speed $n^{2p\over 2+p}$ if $p\in[1,2)$ and speed $n$ if
$p\in[2,\infty]$, and with a certain rate function that also depends on the
parameter $p$; for $p=\infty$, we use the convention $n^{1\over p}=1$. Below
we only discuss the results for $p<\infty$ and refer to [39, Section 8] for a
compilation of results for general product measures satisfying certain tail
conditions (including the uniform measure on ${\mathbb{B}}_{\infty}^{n}$).

####  3.1.1 Annealed LDP – the case $p\in[2,\infty)$

For $p\in[2,\infty)$, denoting by $\mu_{p}\in\mathcal{M}_{1}(\mathbb{R})$ the
$p$-Gaussian measure having Lebesgue density given by (9), we consider the
function

$\Phi_{p}(t_{0},t_{1},t_{2}):=\log\Bigg{(}\int_{\mathbb{R}}\int_{\mathbb{R}}e^{t_{0}z^{2}+t_{1}zy+t_{2}|y|^{p}}\,\mu_{2}(\textup{d}z)\mu_{p}(\textup{d}y)\Bigg{)},\qquad
t_{0},t_{1},t_{2}\in\mathbb{R},$

which satisfies $\Phi_{p}(t_{0},t_{1},t_{2})<\infty$ whenever $t_{0}<{1\over
2}$, $t_{1}\in\mathbb{R}$, and $t_{2}<{1\over p}$. The rate function that
describes the large deviations behavior is given by the Legendre transform of
$\Phi_{p}$, i.e., we define

$\operatorname{\mathbb{I}}_{p}(x):=\inf_{\tau_{0}>0,\tau_{1}\in\mathbb{R},\tau_{2}>0\atop{\tau_{0}^{-1/2}\tau_{1}\tau_{2}^{-1/p}=x}}\Phi^{*}(\tau_{0},\tau_{1},\tau_{2});$

the infimum that appears already hints at an application of the contraction
principle, i.e., the desired LDP will be contracted from another one. Let us
now state the LDP for random one-dimensional projections corresponding to [39,
Theorem 2.2].

######  Theorem A.

Let $p\in[2,\infty)$. Then the sequence of random variables
$(W^{(n,p)})_{n\in\mathbb{N}}$ satisfies an LDP on $\mathbb{R}$ with speed $n$
and quasi-convex, symmetric, good rate function
$\operatorname{\mathbb{I}}_{p}$.

###### Idea of Proof..

First of all, for the simple and not quite so interesting case $p=2$, we refer
to [39, Theorem 2.12]; the rate function is then given by
$x\mapsto-\frac{1}{2}\log(1-x^{2})$ for $x\in(-1,1)$ and is $\infty$
otherwise.

So assume now that $p\in(2,\infty)$. Using the specific form of the
$p$-Gaussian measures (see (9)), one easily checks that for $t_{0}<{1\over
2}$, $t_{1}\in\mathbb{R}$, and $t_{2}<{1\over p}$,

$\Phi_{p}(t_{0},t_{1},t_{2})=-\frac{1}{p}\log(1-pt_{2})-\frac{1}{2}\log(1-2t_{0})+\log\Bigg{(}\int_{\mathbb{R}}e^{\frac{1}{2}t_{1}^{2}(1-pt_{2})^{-2/p}(1-2t_{0})^{-1}y^{2}}\,\mu_{p}(\textup{d}y)\Bigg{)},$

which is finite, since $p>2$. Thus, the effective domain
$\mathscr{D}_{\Phi_{p}}=\\{z\in\mathbb{R}^{3}\,:\,\Phi_{p}(z)<\infty\\}$ of
$\Phi_{p}$ satisfies

$\mathscr{D}_{\Phi_{p}}^{\circ}=\Big{(}-\infty,\frac{1}{2}\Big{)}\times\mathbb{R}\times\Big{(}-\infty,\frac{1}{p}\Big{)}\ni
0.$

It now follows from a general version of Cramér’s theorem [27, Theorem 2.2.30]
(which applies to the multivariate setting if $0$ is an interior point of the
effective domain of the cumulant generating function of the considered random
vectors) that the sequence

$S^{(n,p)}:=\frac{1}{n}\sum_{i=1}^{n}\big{(}|Z_{i}^{(n)}|^{2},Y_{i}^{(n,p)}Z_{i}^{(n)},|Y_{i}^{(n,p)}|^{p}\big{)},\quad
n\in\mathbb{N},$

with

$Y^{(n,p)}=(Y^{(n,p)}_{1},\dots,Y^{(n,p)}_{n})\text{ having iid coordinates
with common distribution $\mu_{p}$}$

and

$Z^{(n)}=(Z^{(n)}_{1},\dots,Z^{(n)}_{n})\text{ being a vector of independent
standard Gaussians},$

satisfies an LDP in $\mathbb{R}^{3}$ with speed $n$ and good rate function
given by the Legendre transform $\Phi^{*}$; note that $\Phi_{p}$ is the log-
moment generating function of the summand in the definition of $S^{(n,p)}$.
The desired LDP will now be contracted from the latter one via the continuous
mapping $T_{p}:(0,\infty)\times\mathbb{R}\to\mathbb{R}$,
$T_{p}(\tau_{0},\tau_{1},\tau_{2}):=\tau_{0}^{-1/2}\tau_{1}\tau_{2}^{-1/p}$.
More precisely, one readily checks that

$T(S^{(n,p)})=\frac{n^{1/p}}{n^{1/2}}\frac{\sum_{i=1}^{n}Y^{(n,p)}_{i}Z_{i}^{(n)}}{\|Y^{(n,p)}\|_{p}\|Z^{(n)}\|_{2}},$

and so one obtains an LDP for the sequence $T(S^{(n,p)})$, $n\in\mathbb{N}$,
with speed $n$ and rate function $\operatorname{\mathbb{I}}_{p}$. Now it is
only left to show that the LDP for the latter sequence carries over (with
identical speed and rate) to our sequence $W^{(n,p)}$, $n\in\mathbb{N}$. For
this we refer to [39, Lemma 3.4]. ∎

######  Remark 3.1.

We refer the reader to [3, Theorem 1.1] for the case of high-dimensional
projections discussed in the introduction.

####  3.1.2 Annealed LDP – the case $p\in[1,2)$

As we shall see now, when we are in the case $p\in[1,2)$, the random
projections display a completely different large deviations behavior. For
$p\in[1,2)$, we define the function $\mathbb{J}_{p}:\mathbb{R}\to[0,\infty)$
via

$\mathbb{J}_{p}(x):=\frac{1}{r_{p}}|x|^{r_{p}},$

with $r_{p}:=\frac{2p}{2+p}$. It is important to note that $r_{p}<1$ for $p<2$
and so the following LDP occurs at a different speed $n^{r_{p}}$, which is
slower than the speed $n$ in the LDP for $p\geq 2$. The following result
corresponds to [39, Theorem 2.3].

######  Theorem B.

Let $p\in[1,2)$. Then the sequence of random variables
$(W^{(n,p)})_{n\in\mathbb{N}}$ satisfies an LDP on $\mathbb{R}$ with speed
$n^{r_{p}}$ and quasi-convex, symmetric, good rate function $\mathbb{J}_{p}$.

It is immediately clear that this result cannot be proved in the same way as
the LDP in Theorem A, because for $p<2$, $\Phi_{p}(t_{0},t_{1},t_{2})=\infty$
for $t_{1}\neq 0$; this already suggests that the LDP, if it exists, occurs at
a different speed slower than $n$.

###### Idea of Proof of Theorem B..

In a first step, one proves that if $Y\sim\mu_{p}$ and $Z\sim\mu_{2}$, and $Y$
is independent of $Z$, then it holds that

$\lim_{t\to\infty}\frac{1}{t^{r_{p}}}\log\big{(}\mathbb{P}[YZ\geq
t]\big{)}=-\frac{1}{r_{p}};$ (10)

to prove this, one uses the $p$-Gaussian tail bounds saying that, for any
$x\geq 0$,

$\frac{x}{x^{p}+1}e^{-x^{p}/p}\leq\int_{x}^{\infty}e^{-y^{p}/p}\,\textup{d}y\leq\frac{1}{x^{p-1}}e^{-x^{p}/p}.$
(11)

In the second step, one uses (10) together wit a result of M. A. Arcones on
large deviations of empirical processes with non-standard rates [8, Theorem
2.1] (alternatively one may use [40, Theorem 1] on LDPs for random variables
with stretched exponential tails) to show that the random sequence

$V^{(n,p)}:=\frac{1}{n}\sum_{i=1}^{n}Y_{i}^{(n,p)}Z_{i}^{(n)},\quad
n\in\mathbb{N},$

with $Y^{(n,p)}=(Y^{(n,p)}_{1},\dots,Y^{(n,p)}_{n})$ having iid coordinates
with distribution $\mu_{p}$ and $Z^{(n)}=(Z^{(n)}_{1},\dots,Z^{(n)}_{n})$
being a vector of independent standard Gaussians, satisfies an LDP with speed
$n^{r_{p}}$ and quasi-convex good rate function $\mathbb{J}_{p}$.

In a third step, one proves that the sequence

$\frac{n^{1/p}}{n^{1/2}}\frac{\sum_{i=1}^{n}Y^{(n,p)}_{i}Z_{i}^{(n)}}{\|Y^{(n,p)}\|_{p}\|Z^{(n)}\|_{2}},\quad
n\in\mathbb{N},$

which we considered above in the proof of Theorem A, and $V^{(n,p)}$,
$n\in\mathbb{N}$, are exponentially equivalent with speed $n^{r_{p}}$; thus
they satisfy the same LDP. Again, as mentioned in the proof of Theorem A, the
LDP now carries over to $(W^{(n,p)})_{n\in\mathbb{N}}$ with speed $n^{r_{p}}$
and quasi-convex, symmetric, good rate function $\mathbb{J}_{p}$ (see [39,
Lemma 3.4]). ∎

######  Remark 3.2.

Again, we refer the reader to [3, Theorem 1.2] for the case of high-
dimensional projections, which shall not be discussed here.

###  3.2 Large deviations for $\ell_{q}$-norms of random vectors in
$\ell_{p}$-balls

We now present some LDPs obtained by Z. Kabluchko, J. Prochno, and C. Thäle
for $\ell_{q}^{n}$-norms of random vectors chosen uniformly at random from
$\ell_{p}^{n}$-balls [63]; we shall skip the central and non-central limit
theorems the authors obtained and focus on the large deviations part. As a
motivation, let us start with the classical deviation results of G. Schechtman
and J. Zinn [107, Theorem 3] (see [107, Corollary 4] for the normalized
Lebesgue measure) and of A. Naor in [89, Theorem 2]. The authors proved that
if $1\leq p<q<\infty$, then

$\mathbb{P}\big{[}n^{1/p-1/q}\|Z\|_{q}>z\big{]}\leq e^{-cn^{p/q}z^{p}}$ (12)

for all $z>T(p,q)$, with $c:=1/T(p,q)$. Here $Z$ can be either uniformly
distributed in ${\mathbb{B}}_{p}^{n}$ or distributed according to the cone
measure on the boundary of ${\mathbb{B}}_{p}^{n}$.

The authors of [63] obtained an LDP counterpart, i.e., an asymptotic version
of (12). In fact, they identify $n^{p/q}$ in the exponent on the right-hand
side of (12) as the speed of the LDP and $z^{p}$ as the asymptotically leading
term of the rate function as $z\to\infty$. This LDP is in a sense optimal, and
one can, contrary to [107], identify the exact constant $1/p$ in the exponent.
It is important to note, however, that in general neither one of the results
implies the other. Only for deviation parameters $z$ from a fixed compact
interval, the result of [63] is optimal and indeed stronger. The following
result is based on large deviations techniques for sums of random variables
with stretched exponential tails obtained in [40] (i.e., a generalized version
of Theorem 2.2 above) and corresponds to [63, Theorem 1.3].

######  Theorem C.

Assume that $1\leq p<\infty$ and, for $n\in\mathbb{N}$, let $Z^{(n,p)}$ be
uniformly distributed on ${\mathbb{B}}_{p}^{n}$. If $p<q<\infty$, then the
sequence $\|{\bf Z}\|:=(n^{1/p-1/q}\|Z^{(n,p)}\|_{q})_{n\in\mathbb{N}}$
satisfies an LDP in $\mathbb{R}$ with speed $n^{p/q}$ and good rate function

$\operatorname{\mathbb{I}}_{\bf\|Z\|}(z):=\begin{cases}{1\over
p}\big{(}z^{q}-M_{p}(q)\big{)}^{p/q}&:z\geq M_{p}(q)^{1/q}\\\
\infty&:\text{otherwise}\,,\end{cases}$

where
$M_{p}(q):=\frac{p^{q/p}}{q+1}\,\frac{\Gamma(1+\frac{q+1}{p})}{\Gamma(1+\frac{1}{p})}$.

###### Idea of Proof..

The case $p<q$ is more delicate than $q\leq p$ because of the lack of finite
exponential moments. Thus, instead of Cramér’s theorem, one uses a version for
random variables with heavy tails. More precisely, the following result due to
N. Gantert, K. Ramanan, and F. Rembart [40, Theorem 1] (which is actually
reformulated in a suitable way here) is used: let $X,X_{1},X_{2},\ldots$ be
non-negative iid random variables, and assume that there are constants
$r\in(0,1)$, $t_{0}>0$, and slowly varying functions
$c_{1},c_{2},b:(0,\infty)\to(0,\infty)$ such that

$c_{1}(t)e^{-b(t)t^{r}}\leq\mathbb{P}[X\geq t]\leq
c_{2}(t)e^{-b(t)t^{r}}\,,\qquad t\geq t_{0}\,.$

Then, the sequence of random variables ${1\over n}\sum\limits_{i=1}^{n}X_{i}$,
$n\in\mathbb{N}$, satisfies an LDP on $\mathbb{R}$ with speed $b(n)n^{r}$ and
good rate function

$\operatorname{\mathbb{I}}(z)=\begin{cases}(x-\mathbb{E}X)^{r}&:z\geq\mathbb{E}X\\\
\infty&:\text{otherwise}\,.\end{cases}$

For each $n\in\mathbb{N}$, let $Y_{1},\ldots,Y_{n}\sim\mu_{p}$ be independent
$p$-generalized Gaussians and define

$S_{n}:=\sum_{i=1}^{n}|Y_{i}|^{q}\,.$

In a first step, the authors apply Cramér’s theorem for stretched exponentials
from [40] to prove an LDP for the random sequence ${\bf
S}:=(S_{n})_{n\in\mathbb{N}}$. Using the $p$-Gaussian tail bounds from (11),
one can check that, for sufficiently large $z\in\mathbb{R}$,

$c_{1}(z)e^{-b(z)z^{p/q}}\leq\mathbb{P}[|Y_{1}|^{q}\geq z]\leq
c_{2}(z)e^{-b(z)z^{p/q}},$

with appropriate slowly varying functions
$c_{1},c_{2},b:(0,\infty)\to(0,\infty)$, where $b(z)$ satisfies
$b(z)\to{1\over p}$, as $z\to\infty$. Thus, $\bf S$ satisfies an LDP on
$\mathbb{R}$ with speed $n^{p/q}$ and good rate function

$\operatorname{\mathbb{I}}_{\bf S}(z)=\begin{cases}{1\over
p}(z-M_{p}(q))^{p/q}&:z\geq M_{p}(q)\\\
+\infty&:\text{otherwise}\,.\end{cases}$

In a second step, one applies the contraction principle of Lemma 2.15 with the
continuous function $F(z)=z^{1/q}$, $z\in\mathbb{R}$, to conclude that
$(F(S_{n}))_{n\in\mathbb{N}}=((S_{n})^{1/q})_{n\in\mathbb{N}}$ satisfies an
LDP on $\mathbb{R}$ with speed $n^{p/q}$ and good rate function
$\operatorname{\mathbb{I}}_{\bf\|Z\|}$ from the statement of the theorem.

In a third and last step, one shows that $((S_{n})^{1/q})_{n\in\mathbb{N}}$ is
exponentially equivalent to the sequence
$(n^{1/p-1/q}\|Z^{(n,p)}\|_{q})_{n\in\mathbb{N}}$ of interest. Here one uses
the Schechtman–Zinn probabilistic representation from (8) to obtain the
distributional identity

$n^{1/p-1/q}\|Z^{(n,p)}\|_{q}\overset{d}{=}U^{1/n}{\Big{(}{1\over
n}\sum\limits_{i=1}^{n}|Y_{i}|^{q}\Big{)}^{1/q}\over\Big{(}{1\over
n}\sum\limits_{i=1}^{n}|Y_{i}|^{p}\Big{)}^{1/p}}\,.$

Now, for fixed $\delta>0$ and $\varepsilon\in(0,1)$, one can show that

$\displaystyle\mathbb{P}\left[\,\left|\Big{(}{1\over
n}\sum\limits_{i=1}^{n}|Y_{i}|^{q}\Big{)}^{1/q}-U^{1/n}{\Big{(}{1\over
n}\sum\limits_{i=1}^{n}|Y_{i}|^{q}\Big{)}^{1/q}\over\Big{(}{1\over
n}\sum\limits_{i=1}^{n}|Y_{i}|^{p}\Big{)}^{1/p}}\right|>\delta\right]$
$\displaystyle\leq\mathbb{P}\bigg{[}\Big{(}{1\over
n}\sum\limits_{i=1}^{n}|Y_{i}|^{q}\Big{)}^{1/q}>{\delta\over\varepsilon}\bigg{]}+\mathbb{P}\big{[}U^{1/n}<\sqrt{1-\varepsilon}\,\big{]}+\mathbb{P}\bigg{[}{1\over
n}\sum\limits_{i=1}^{n}|Y_{i}|^{p}>(1-\varepsilon)^{-p/2}\bigg{]}$
$\displaystyle\hskip
130.88284pt+\mathbb{P}\big{[}U^{1/n}>\sqrt{1+\varepsilon}\,\big{]}+\mathbb{P}\bigg{[}{1\over
n}\sum\limits_{i=1}^{n}|Y_{i}|^{p}<(1+\varepsilon)^{-p/2}\bigg{]}\,.$

From Cramér’s theorem [27, Theorem 2.2.30] and [39, Lemma 3.3] one deduces
that the four last terms decay exponentially with speed $n$, while the
exponential asymptotic of the first term has already been determined above. As
a consequence, and since $p/q<1$, one has that

$\displaystyle\limsup_{n\to\infty}{1\over
n^{p/q}}\log\mathbb{P}\left[\,\left|\Big{(}{1\over
n}\sum\limits_{i=1}^{n}|Y_{i}|^{q}\Big{)}^{1/q}-U^{1/n}{\Big{(}{1\over
n}\sum\limits_{i=1}^{n}|Y_{i}|^{q}\Big{)}^{1/q}\over\Big{(}{1\over
n}\sum\limits_{i=1}^{n}|Y_{i}|^{p}\Big{)}^{1/p}}\right|>\delta\right]$
$\displaystyle\qquad\qquad\leq\limsup_{n\to\infty}{1\over
n^{p/q}}\log\mathbb{P}\bigg{[}\Big{(}{1\over
n}\sum\limits_{i=1}^{n}|Y_{i}|^{q}\Big{)}^{1/q}>{\delta\over\varepsilon}\bigg{]}\,,$

which is $-{1\over p}(({\delta\over\varepsilon})^{q}-M_{p}(q))^{p/q}$ if
$\delta/\varepsilon\geq M_{p}(q)^{1/q}$ and $-\infty$ otherwise. Letting
$\varepsilon\to 0$, one concludes that the above limit is equal to $-\infty$.
Thus, the two sequences $(n^{1/p-1/q}\|Z^{(n,p)}\|_{q})_{n\in\mathbb{N}}$ and
$((S_{n})^{1/q})_{n\in\mathbb{N}}$ are exponentially equivalent, thus obeying
the same LDP (see Lemma 2.17). ∎

Let us now present the LDP in the regime $1\leq q\leq p<\infty$, which
corresponds to [63, Theorem 1.2] . As already pointed out, in this case
exponential moments exist and make the large deviations analysis rather
straight forward, but we will also see that the two regimes exhibit a
different behavior from a rate function and speed point of view. In
particular, the form of the rate function already hints at an application of
the large deviations result for pairs of independent random objects presented
in Lemma 2.16. Let us note that the case $p=q$ is rather immediate (see [63,
Theorem 1.4]) and so below we work with $q<p$; the special case $p=\infty$ is
treated in (see [63, Theorem 1.5]).

######  Theorem D.

Assume that $1\leq p<\infty$ and, for $n\in\mathbb{N}$, let $Z^{(n,p)}$ be
uniformly distributed on ${\mathbb{B}}_{p}^{n}$. If $1\leq q<p$, the sequence
$\|{\bf Z}\|:=(n^{1/p-1/q}\|Z^{(n,p)}\|_{q})_{n\in\mathbb{N}}$ satisfies an
LDP with speed $n$ and good rate function

$\operatorname{\mathbb{J}}_{\|\textbf{Z
}\|}(z):=\begin{cases}\inf\limits_{z=z_{1}z_{2}\atop z_{1},z_{2}\geq
0}[{\operatorname{\mathbb{I}}}_{1}(z_{1})+{\operatorname{\mathbb{I}}}_{2}(z_{2})]&:z\geq
0\\\ \infty&:\text{otherwise}\,.\end{cases}$

Here

${\operatorname{\mathbb{I}}}_{1}(z):=\begin{cases}-\log z&:z\in(0,1]\\\
\infty&:\text{otherwise}\end{cases}\qquad\text{and}\qquad\operatorname{\mathbb{I}}_{2}(z):=\begin{cases}\inf\limits_{x\geq
0,y>0\atop{x^{1/q}y^{-1/p}}=z}\Lambda^{*}(x,y)&:z\geq 0\\\
\infty&:z<0\,,\end{cases}$

where $\Lambda^{*}$ is the Legendre transform of the function

$\Lambda(t_{1},t_{2}):=\log\int_{\mathbb{R}}e^{t_{1}|s|^{q}+t_{2}|s|^{p}}\frac{e^{-|s|^{p}/p}}{2p^{1/p}\Gamma(1+1/p)}\,\mathrm{d}s\,.$

###### Idea of Proof..

First of all, one employs again the Schechtmann–Zinn probabilistic
representation from (8) to obtain, for each $n\in\mathbb{N}$,

$n^{1/p-1/q}\|Z^{(n,p)}\|_{q}\stackrel{{\scriptstyle\text{d}}}{{=}}n^{1/p-1/q}U^{1/n}\frac{\|Y^{(n,p)}\|_{q}}{\|Y^{(n,p)}\|_{p}},$

where $Y^{(n,p)}=(Y^{(n,p)}_{1},\ldots,Y^{(n,p)}_{n})$ is a vector of iid
$p$-generalized Gaussians, and $U$ is an independent random variable uniformly
distributed on $[0,1]$. Then, for $n\in\mathbb{N}$, one defines the random
quantities

$W_{n}:=n^{1/p-1/q}\frac{\|Y^{(n,p)}\|_{q}}{\|Y^{(n,p)}\|_{p}}\qquad\text{and}\qquad
S_{n}:=\frac{1}{n}\sum_{i=1}^{n}\big{(}|Y_{i}^{(n,p)}|^{q},|Y_{i}^{(n,p)}|^{p}\big{)}\,.$

Now, for $t=(t_{1},t_{2})\in\mathbb{R}^{2}$,

$\Lambda(t_{1},t_{2})=\log\Big{(}\mathbb{E}\Big{[}e^{\langle
t,(|Y_{1}^{(n,p)}|^{q},|Y_{1}^{(n,p)}|^{p})\rangle}\Big{]}\Big{)}=\log\int_{0}^{\infty}e^{t_{1}s^{q}+(t_{2}-1/p)s^{p}}\frac{\,\mathrm{d}s}{2p^{1/p}\Gamma(1+1/p)}$

is the log-moment generating function of $S_{n}$. Its effective domain is
$\mathbb{R}\times(-\infty,1/p)$ if $q<p$ and
$\\{(t_{1},t_{2})\in\mathbb{R}^{2}\,:\,t_{1}+t_{2}<1/p\\}$ for $q=p$. Thus, by
Cramér’s theorem [27, Theorem 2.2.30], the sequence ${\bf
S}:=(S_{n})_{n\in\mathbb{N}}$ satisfies an LDP in $\mathbb{R}^{2}$ with speed
$n$ and good rate function $\Lambda^{*}$ being the Legendre transform of
$\Lambda$. One can check that $\Lambda^{*}(t_{1},t_{2})=+\infty$ if $t_{1}\leq
0$ or $t_{2}\leq 0$. This implies that the sequence ${\bf S}$ also satisfies
an LDP on $[0,\infty)\times(0,\infty)$ with the same good rate function
$\Lambda^{*}$.

In a next step, one defines the continuous function

$F:[0,\infty)\times(0,\infty)\to\mathbb{R},\quad(x,y)\mapsto x^{1/q}y^{-1/p},$

and notes that, for each $n\in\mathbb{N}$, $W_{n}\overset{d}{=}F(S_{n})$. The
contraction principle of Lemma 2.15 implies that the random sequence ${\bf
W}:=(W_{n})_{n\in\mathbb{N}}$ satisfies an LDP on $\mathbb{R}$ with speed $n$
and good rate function

$\operatorname{\mathbb{I}}_{\bf W}(z)=\begin{cases}\inf\limits_{x\geq
0,y>0\atop{x^{1/q}y^{-1/p}}=z}\Lambda^{*}(x,y)&:z\geq 0\\\
+\infty&:z<0\,.\end{cases}$

In the third and last step, one considers the random variables
$V_{n}:=(U^{1/n},W_{n})$, $n\in\mathbb{N}$, and uses again the fact (see [39,
Lemma 3.3]) that the sequence ${\bf U}:=(U^{1/n})_{n\in\mathbb{N}}$ satisfies
an LDP on $\mathbb{R}$ with speed $n$ and rate function

$\operatorname{\mathbb{I}}_{\bf U}(z)=\begin{cases}-\log z&:z\in(0,1]\\\
\infty&:\text{otherwise}\,.\end{cases}$

Now, since $U^{1/n}$ and $W_{n}$ are independent, the sequence ${\bf
V}:=(V_{n})_{n\in\mathbb{N}}$ satisfies an LDP on $\mathbb{R}^{2}$ with speed
$n$ and good rate function

$\operatorname{\mathbb{I}}_{\bf
V}(z_{1},z_{2}):=\operatorname{\mathbb{I}}_{\bf
U}(z_{1})+\operatorname{\mathbb{I}}_{\bf
W}(z_{2})\,,\qquad(z_{1},z_{2})\in\mathbb{R}^{2}\,,$

by Lemma 2.16 (see also [8, Lemma 2.2]). Applying once more the contraction
principle, but this time with the continuous function
$F:\mathbb{R}^{2}\to\mathbb{R}$, $(x,y)\mapsto xy$, one concludes that the
sequence $U^{1/n}W_{n}\overset{d}{=}n^{1/p-1/q}\|Z^{(n,p)}\|_{q}$,
$n\in\mathbb{N}$, satisfies an LDP on $\mathbb{R}$ with speed $n$ and good
rate function

$\operatorname{\mathbb{J}}_{\|\textbf{Z
}\|}(z)=\inf_{z=z_{1}z_{2}}{\operatorname{\mathbb{I}}}_{\bf
V}(z_{1},z_{2})=\begin{cases}\inf\limits_{z=z_{1}z_{2}\atop z_{1},z_{2}\geq
0}{\operatorname{\mathbb{I}}}_{\bf V}(z_{1},z_{2})&:z\geq 0\\\
\infty&:\text{otherwise}.\end{cases}$

This completes the argument. ∎

###  3.3 Moderate deviations for $\ell_{q}$-norms of random vectors in
$\ell_{p}$-balls

In [67], Z. Kabluchko, J. Prochno, and C. Thäle complemented the LDPs
presented above by MDPs (which occur on scalings between the one of the CLT
and LDP). More precisely, we refer the reader to [67, Theorem C], where the
result is actually obtained for more general distributions on the
$\ell_{p}^{n}$-balls as introduced by F. Barthe, O. Guédon, S. Mendelson, and
A. Naor in [16]. We shall not sketch the proof of the MDP and instead refer to
[67] directly.

Let us recall that an MDP is formally nothing else than an LDP, but with
important differences in the behavior of the two principles. For instance,
while LDPs provide estimates on the scale of a law of large numbers, MDPs
describe the probabilities at scales between a law of large numbers and a
distributional limit theorem (like a CLT). Moreover, while the rate function
in an LDP depends in a subtle way on the distribution of the underlying random
variables, the rate function in an MDP, in typical situations, is only
parametric in the variance that occurs and is inherited from a CLT. Let us
recall that a sequence $(X_{n})_{n\in\mathbb{N}}$ of random vectors in
$\mathbb{R}^{d}$ ($d\in\mathbb{N}$) satisfies an LDP with _s_ peed $s_{n}$ and
good rate function $\operatorname{\mathbb{I}}:\mathbb{R}^{d}\to[0,\infty]$ if
and only if

$\begin{split}-\inf_{x\in
A^{\circ}}\operatorname{\mathbb{I}}(x)&\leq\liminf_{n\to\infty}s_{n}^{-1}\log\mathbb{P}[X_{n}\in
A]\leq\limsup_{n\to\infty}s_{n}^{-1}\log\mathbb{P}[X_{n}\in
A]\leq-\inf_{x\in\overline{A}}\operatorname{\mathbb{I}}(x)\end{split}$

for all measurable $A\subset\mathbb{R}^{d}$. One says that a sequence
$(X_{n})_{n\in\mathbb{N}}$ satisfies an MDP if the speed sequence
$(s_{n})_{n\in\mathbb{N}}$ is given by $s_{n}=b_{n}\sqrt{n}$ with a positive
sequence $(b_{n})_{n\in\mathbb{N}}$ satisfying $b_{n}=\omega(1)$ and
$b_{n}=o(\sqrt{n})$, where for two sequences $(x_{n})_{n\in\mathbb{N}}$ and
$(y_{n})_{n\in\mathbb{N}}$ we use the Landau notation $x_{n}=o(y_{n})$ if
$\lim_{n\to\infty}\frac{x_{n}}{y_{n}}=0$ and $x_{n}=\omega(y_{n})$ if
$\lim_{n\to\infty}|\frac{x_{n}}{y_{n}}|=\infty$. We refer the reader to [27,
Chapter 3.7] and especially part (c) of the remark after Theorem 3.7.1 for
further details on terminology.

######  Theorem E.

Let $0<p<\infty$ and $0<q<\infty$ with $q<p$. Let $(b_{n})_{n\in\mathbb{N}}$
be a sequence of positive real numbers satisfying $b_{n}=\omega(1)$ and
$b_{n}=o(\sqrt{n})$. For each $n\in\mathbb{N}$, let $Z^{(n,p)}$ be uniformly
distributed on ${\mathbb{B}}_{p}^{n}$. Then the sequence of random variables

$\displaystyle{\sqrt{n}\over b_{n}}\bigg{(}{n^{{1/p}-{1/q}}\over
M_{p}(q)^{1/q}}\|Z^{(n,p)}\|_{q}-1\bigg{)},\qquad n\in\mathbb{N},$

satisfies an MDP with speed $b_{n}^{2}$ and good rate function
$\operatorname{\mathbb{I}}(t)={t^{2}\over 2\sigma^{2}}$, $t\in\mathbb{R}$,
where

$\sigma^{2}:={1\over q^{2}}\bigg{(}{\Gamma({1\over p})\Gamma({2q+1\over
p})\over\Gamma({q+1\over p})^{2}}-1\bigg{)}-{1\over p}\qquad\text{and}\qquad
M_{p}(q):=\frac{p^{q/p}}{q+1}\,\frac{\Gamma(1+\frac{q+1}{p})}{\Gamma(1+\frac{1}{p})}.$

In particular, Theorem E implies that, for all $t\in\mathbb{R}$,

$\lim_{n\to\infty}{1\over b_{n}^{2}}\log\mathbb{P}\Bigg{[}{\sqrt{n}\over
b_{n}}\bigg{(}{n^{{1/p}-{1/q}}\over
M_{p}(q)^{1/q}}\|Z^{(n,p)}\|_{q}-1\bigg{)}\geq t\Bigg{]}=-{t^{2}\over
2\sigma^{2}}.$

######  Remark 3.3.

Note that already for $p=q$ the core term leading the MDP vanishes and
therefore, one does not obtain an MDP with a non-trivial rate function.

###  3.4 Large deviations, moderate deviations, and the KLS conjecture

Arguably, the major open problem in asymptotic geometric analysis is the
Kannan–Lovász–Simonovits (KLS) conjecture. Its origin lies in theoretical
computer science, where it arose in the study of sampling algorithms for high-
dimensional convex bodies. A specific situation of interest is to design an
algorithm efficiently computing the volume of a $d$-dimensional convex body.
Such an algorithm is given a convex body $K\subseteq\mathbb{R}^{d}$ and a
quality parameter $\varepsilon\in(0,\infty)$. The body $K$ is represented by a
membership oracle, which, for each $x\in\mathbb{R}^{d}$, can decide whether or
not $x\in K$. The algorithm returns a number
$V:=V(K,\varepsilon)\in\mathbb{R}$ such that
$(1-\varepsilon)\mathrm{vol}_{d}(K)\leq
V\leq(1+\varepsilon)\mathrm{vol}_{d}(K)$. The efficiency of the designed
algorithm is measured in terms of the number of arithmetic operations and
calls to the membership oracle.

While it is known through the works of I. Bárány and Z. Füredi [14] and of G.
Elekes [32] that deterministic algorithms are inefficient, there are
reasonable randomized algorithms that can accurately compute the volume of
convex bodies with high probability in polynomial time (see, e.g., the work of
M. Dyer, A. Frieze, and R. Kannan [30]). The construction of this randomized
algorithm is connected to an isoperimetric inequality for log-concave
probability measures on $\mathbb{R}^{d}$ (see [1] for an introduction to the
KLS conjecture and [113] for a detailed explanation on its connection with
problems in computer science). The constant appearing in this isoperimetric
inequality, referred to as Cheeger’s constant, is of particular interest and
directly linked to the KLS conjecture, which can be stated as follows (see
[20, Theorem 1.1]):

KLS Conjecture. There exists an absolute constant $C\in(0,\infty)$ such that
for all $d\in\mathbb{N}$, every centered random vector $X$ with log-concave
distribution, and any locally Lipschitz function
$f:\mathbb{R}^{d}\to\mathbb{R}$ such that $f(X)$ is of finite variance,

$\mathrm{Var}[f(X)]\leq C\,\lambda_{X}^{2}\,\mathbb{E}\big{[}\|\nabla
f(X)\|_{2}^{2}\big{]},$

where $\lambda_{X}^{2}:=\sup_{\theta\in{\mathbb{S}}^{d-1}}\mathbb{E}\langle
X,\theta\rangle^{2}$.

Using the localization lemma, R. Kannan, L. Lovász, and M. Simonovits proved
the conjecture with a factor $(\mathbb{E}\|X\|_{2})^{2}$ instead of
$\lambda_{X}^{2}$ (see [69]), and later S. Bobkov [19] improved this estimate.
The currently best known bound has recently been obtained by B. Klartag [78],
who showed that the conjecture in $\mathbb{R}^{d}$ holds true up to a factor
of $\sqrt{\log d}$.

Even though the bound has been improved considerably in recent years, it is
not clear whether the conjecture is indeed true. In [5], D. Alonso-Gutiérrez,
J. Prochno, and C. Thäle established a new connection between the KLS
conjecture and the study of LDPs and MDPs for isotropic log-concave random
vectors, providing a potential route to disproving the conjecture.

As mentioned before, an MDP is nothing else than an LDP for a differently
normalized sequence of random variables, i.e., if a sequence
$(t_{n}^{-1}X_{n})_{n\in\mathbb{N}}$ of random variables satisfies an LDP with
some speed and some rate function, an LDP for the sequence of random variables
$\tilde{t}_{n}^{-1}X_{n}$ with $\tilde{t}_{n}\to\infty$ and
$\tilde{t}_{n}=o(t_{n})$ is referred to as an MDP.

The following result corresponds to [5, Theorem A] and establishes a
connection between the KLS conjecture and moderate/large deviations
principles. Essentially, it is a consequence of the fact, known from [1, 46,
83], that the KLS conjecture would imply a sharp concentration bound for the
Euclidean norm of isotropic log-concave random vectors. Below, for two
positive sequences of quantities, we write $\approx$ to denote equivalence up
to absolute constants.

######  Theorem F.

Let $1\leq k_{n}\leq n$ be a sequence of integers such that $k_{n}=\omega(1)$
and let $(\xi_{n})_{n=1}^{\infty}$ be a sequence of isotropic log-concave
random vectors in $\mathbb{R}^{k_{n}}$. Consider a sequence of random
variables $X_{n}=\frac{\|\xi_{n}\|_{2}}{\sqrt{k_{n}}}$, $n\in\mathbb{N}$,
which satisfies (7) with speed $s_{n}$ and rate function
$\operatorname{\mathbb{I}}$. Assume that one of the following two conditions
is satisfied:

  * (a)

$s_{n}=o(\sqrt{k_{n}})$ and $\operatorname{\mathbb{I}}$ is non-singular, i.e.,
$\operatorname{\mathbb{I}}(x)\neq\operatorname{\mathbb{I}}_{0}(x):=\begin{cases}0&:x=1\\\
\infty&:\text{otherwise}.\end{cases}$

  * (b)

$s_{n}\approx\sqrt{k_{n}}$ and
$\inf\limits_{t>t_{0}}\frac{\inf_{x\in(t,\infty)}\operatorname{\mathbb{I}}(x)}{t}=0$
for some absolute constant $t_{0}\in(1,\infty)$.

Then the KLS conjecture is false.

For a sequence of random vectors $\xi_{n}\in\mathbb{R}^{k_{n}}$,
$n\in\mathbb{N}$, as given in Theorem F, it is well known that
$\mathbb{E}\|\xi_{n}\|_{2}/\sqrt{k_{n}}\to 1$, as $n\to\infty$. Against this
light, the natural scale for a law of large numbers, and hence an LDP, is
$\sqrt{k_{n}}$, while scales with $t_{n}=o(\sqrt{k_{n}})$ and
$t_{n}=\omega(1)$ correspond to an MDP. Thus, Theorem F establishes a
connection between LDPs or MDPs for isotropic log-concave random vectors and
the KLS conjecture, showing that by constructing a sequence of random vectors
$\xi_{n}$, $n\in\mathbb{N}$, such that $X_{n}$, $n\in\mathbb{N}$, follows an
LDP or MDP with an appropriate speed and/or rate function would disprove the
variance and hence the KLS conjecture. Moreover, the authors show that
(rescaled) crosspolytopes constitute a critical case in Theorem F as they just
fail to satisfy condition (b) there.

######  Remark 3.4.

Beyond the connection to the KLS conjecture, the authors of [5] prove a number
of other results (see [5, Theorems B and C]), more precisely, they study
moderate deviations for the Euclidean norm of random orthogonally projected
random vectors in $\ell_{p}^{d}$-balls. This leads to a number of interesting
observations: (1) for $p\geq 2$ the rate function in the MDP undergoes a phase
transition, depending on whether the scaling is below the square-root of the
subspace dimensions or comparable; (2) for $1\leq p<2$ and comparable subspace
dimensions, the rate function again displays a phase transition depending on
its growth relative to $d^{p/2}$.

###### Idea of Proof of Theorem F..

First, we recall the following fact (see [1, Theorem 1.15], [46] or [83]: if
the KLS conjecture were true, then there would exist an absolute constant
$C\in(0,\infty)$ such that, for every $n\in\mathbb{N}$ and all $t>0$,

$\mathbb{P}\left[\left|\frac{\|\xi_{n}\|_{2}}{\sqrt{k_{n}}}-1\right|>t\right]\leq
2e^{-Ct\sqrt{k_{n}}}.$

But this would mean that

$\displaystyle\log\mathbb{P}\left[\frac{\|\xi_{n}\|_{2}}{\sqrt{k_{n}}}-1>t\right]$
$\displaystyle\leq$
$\displaystyle\log\mathbb{P}\left[\left|\frac{\|\xi_{n}\|_{2}}{\sqrt{k_{n}}}-1\right|>t\right]\leq\log
2-Ct\sqrt{k_{n}}$

and

$\displaystyle\log\mathbb{P}\left[\frac{\|\xi_{n}\|_{2}}{\sqrt{k_{n}}}<1-t\right]$
$\displaystyle\leq$
$\displaystyle\log\mathbb{P}\left[\left|\frac{\|\xi_{n}\|_{2}}{\sqrt{k_{n}}}-1\right|>t\right]\leq\log
2-Ct\sqrt{k_{n}},$

and thus, for every $n\in\mathbb{N}$ and $t>0$,

$\displaystyle\frac{\log\mathbb{P}\left[\frac{\|\xi_{n}\|_{2}}{\sqrt{k_{n}}}>1+t\right]}{s_{n}}$
$\displaystyle\leq$ $\displaystyle\frac{\log
2}{s_{n}}-\frac{Ct\sqrt{k_{n}}}{s_{n}}$

and

$\displaystyle\frac{\log\mathbb{P}\left[\frac{\|\xi_{n}\|_{2}}{\sqrt{k_{n}}}<1-t\right]}{s_{n}}$
$\displaystyle\leq$ $\displaystyle\frac{\log
2}{s_{n}}-\frac{Ct\sqrt{k_{n}}}{s_{n}}.$

Taking the limit inferior as $n\to\infty$, and using the assumption that
$\sqrt{k_{n}}^{-1}\|X_{n}\|_{2}$ satisfies (7) with speed $s_{n}$ and rate
function $\operatorname{\mathbb{I}}$, one makes the following observation: if
$s_{n}=o(\sqrt{k_{n}})$, then, for every $t>0$,

$-\inf_{x\in(1+t,\infty)}\operatorname{\mathbb{I}}(x)\leq-\infty\quad\text{and}\quad-\inf_{x\in(-\infty,1-t)}\operatorname{\mathbb{I}}(x)\leq-\infty,$

which implies that $\operatorname{\mathbb{I}}$ is identically equal to
$\infty$ on $\mathbb{R}\setminus\\{1\\}$. Since

$\mathbb{P}\left[\frac{\|\xi_{n}\|_{2}}{\sqrt{k_{n}}}\in\mathbb{R}\right]=1,$

one has that, for every $n\in\mathbb{N}$,

$\frac{\log\mathbb{P}\left[\frac{\|\xi_{n}\|_{2}}{\sqrt{k_{n}}}\in\mathbb{R}\right]}{s_{n}}=0.$

Hence, taking the limit as $n\to\infty$,

$0=-\inf_{x\in\mathbb{R}}\operatorname{\mathbb{I}}(x)=-\operatorname{\mathbb{I}}(1),$

which implies that $\operatorname{\mathbb{I}}$ would coincide with the
singular rate function $\operatorname{\mathbb{I}}_{0}$, a contradiction to the
assumption that $\operatorname{\mathbb{I}}\neq\operatorname{\mathbb{I}}_{0}$.

If otherwise $s_{n}\approx\sqrt{k_{n}}$, then, for any $t>0$, there exists a
constant $C_{1}\in(0,\infty)$ such that

$-\inf_{x\in(t+1,\infty)}\operatorname{\mathbb{I}}(x)\leq-C_{1}t,$

which implies that

$\frac{\inf_{x\in(t+1,\infty)}\operatorname{\mathbb{I}}(x)}{t+1}>C_{1}{t\over
t+1}$

for all $t>0$. This is equivalent to the fact that, for every $t>1$,

$\frac{\inf_{x\in(t,\infty)}\operatorname{\mathbb{I}}(x)}{t}>C_{1}{t-1\over
t}.$

Now, for every $t_{0}>1$, one has that, for all $t\in[t_{0},\infty)$,
$C_{1}{t-1\over t}>C_{2}(t_{0})$. Thus, for any $t_{0}>1$,

$\inf_{t>t_{0}}\frac{\inf_{x\in(t,\infty)}\operatorname{\mathbb{I}}(x)}{t}\geq{C_{2}(t_{0})}.$

However, this is a contradiction to the assumption. ∎

##  4 Large deviations results and techniques in the world of Schatten classes

The Schatten $p$-class $\mathcal{S}_{p}$ ($0<p\leq\infty$) is the collection
of all compact operators between Hilbert spaces for which the sequence of
their singular values belongs to the sequence space $\ell_{p}$, including the
important cases of the nuclear or trace class operators ($p=1$) and
Hilbert–Schmidt operators ($p=2$). Having been introduced by R. Schatten in
[103, Chapter 6, pp. 71], back then referring to a compact operator as a
completely continuous one, they are among the most prominent unitary operator
ideals studied in functional analysis today. In fact, he worked in the more
general setting of symmetric gauge functions and the unitarily invariant
crossnorms on the subalgebra of finite rank operators generated by them. The
very origin of Schatten’s work can be traced back to his paper [102] and the
subsequent works [104, 105] with J. von Neumann, studying nuclear operators on
Hilbert spaces; on Banach spaces this had later been considered by A. F.
Ruston [100] and on locally convex spaces by A. Grothendieck [48]. Before
Schatten’s monograph appeared, _spaces of completely continuous operators_ had
received comparably little attention in the literature. Today, Schatten
classes provide the mathematical framework for modern applied mathematics
around low-rank matrix recovery (the non-commutative analogue to the classical
compressed sensing approach) and completion (see, e.g., [23, 24, 36, 81, 99]),
which is only one reason for the increased interest in their structure in
recent years. The Schatten class $\mathcal{S}_{p}$ may be considered a non-
commutative version of the classical $\ell_{p}$ sequence space and both share
various structural characteristics, but while there are several similarities
on different levels, there are also many differences in their analytic,
geometric, and probabilistic behavior. In fact, there are several situations
in which the matrix spaces are harder to handle and where arguments are
considerably more delicate and complicated.

From both the local and the global point of view the study of Schatten classes
has a long tradition in geometric functional analysis and their structure has
been investigated intensively in the past 50 years. For instance, Y. Gordon
and D. R. Lewis proved that $\mathcal{S}_{p}$ for $p\neq 2$ fails to have
local unconditional structure and therefore does not have an unconditional
basis [45]. This answered a question of S. Kwapień and A. Pełczyński, who had
previously shown in [82] that the Schatten trace class $\mathcal{S}_{1}$
(naturally identified with $\ell_{2}\otimes_{\pi}\ell_{2}$) as well as
$\mathcal{S}_{\infty}$ are not isomorphic to subspaces with an unconditional
basis. In 1974, N. Tomczak-Jaegermann succeeded in [109] to prove that
$\mathcal{S}_{1}$ has Rademacher cotype $2$. More recently, H. König, M.
Meyer, and A. Pajor obtained in [79] that the isotropic constants of the unit
balls in $\mathcal{S}_{p}^{N}$ are bounded above by absolute constants for all
$1\leq p\leq\infty$. The concentration of mass properties of unit balls in the
Schatten $p$-classes were studied by O. Guédon and G. Paouris in [49] and J.
Radke and B.-H. Vritsiou were able to prove the thin-shell conjecture for
$\mathcal{S}_{\infty}^{N}$ [97]. In [53], A. Hinrichs, J. Prochno, and J.
Vybíral succeeded in determining the asymptotic behavior of entropy numbers
for natural identities $\mathcal{S}_{p}^{N}\hookrightarrow\mathcal{S}_{q}^{N}$
up to absolute constants for all $0<p,q\leq\infty$ and in [54] studied the
sequences of Gelfand numbers for natural embeddings
$\mathcal{S}_{p}^{N}\hookrightarrow\mathcal{S}_{q}^{N}$ of Schatten classes,
providing asymptotically sharp bounds (up to constants depending on $p$ and
$q$) for almost all regimes, thereby extending classical results for finite-
dimensional $\ell_{p}$ sequence spaces by E. D. Gluskin [43, 44] and A. Y.
Garnaev and E. D. Gluskin [41] to the non-commutative setting. Their results
have then been complemented by J. Prochno and M. Strzelecki in [94]. In a
series of papers, Z. Kabluchko, J. Prochno, and C. Thäle computed the precise
asymptotic volume and the volume ratio for Schatten $p$-classes for
$0<p\leq\infty$ [64] and proved a Schechtman–Schmuckenschläger type result for
the volume of intersections of unit balls [65].

Given that there had already been obtained a number of large deviations
results for $\ell_{p}^{n}$-spaces, it was only natural from the functional
analytic point of view to study LDPs in the non-commutative setting. This was
done by Z. Kabluchko, J. Prochno, and C. Thäle, who obtained Sanov-type large
deviations for the empirical spectral measures of random matrices in Schatten
unit balls [66]. In this section, we shall focus on the results obtained
there.

###  4.1 Self-adjoint and non self-adjoint Schatten $p$-classes

Let us describe the general setting, going beyond the mere case of Schatten
$p$-classes, in which the results of [66] were obtained.

Consider $\beta\in\\{1,2,4\\}$ and let $\mathscr{H}_{n}(\mathbb{F}_{\beta})$
be the collection of all self-adjoint $(n\times n)$-matrices with entries from
the skew field $\mathbb{F}_{\beta}$, where $\mathbb{F}_{1}=\mathbb{R}$,
$\mathbb{F}_{2}=\mathbb{C}$, or $\mathbb{F}_{4}=\mathbb{H}$, the set of
Hamiltonian quaternions. The standard Gaussian distribution on
$\mathscr{H}_{n}(\mathbb{F}_{\beta})$ is known as the GOE (Gaussian orthogonal
ensemble) if $\beta=1$, the GUE (Gaussian unitary ensemble) if $\beta=2$, and
the GSE (Gaussian symplectic ensemble) if $\beta=4$. By
$\lambda_{1}(A),\ldots,\lambda_{n}(A)$ we shall denote the (real) eigenvalues
of a matrix $A$ from $\mathscr{H}_{n}(\mathbb{F}_{\beta})$ and consider the
following Schatten-type unit ball, which can be regarded as a matrix analogue
to the classical $\ell_{p}^{n}$-balls and is defined as

${\mathbb{B}}_{p,\beta}^{n}:=\left\\{A\in\mathscr{H}_{n}(\mathbb{F}_{\beta}):\sum_{j=1}^{n}|\lambda_{j}(A)|^{p}\leq
1\right\\},\qquad\beta\in\\{1,2,4\\}\quad\text{and}\quad 0<p\leq\infty.$

If $p=\infty$, then the sum in the definition is replaced by
$\max\\{|\lambda_{j}(A)|:j=1,\ldots,n\\}$ . The boundary of the matrix ball
${\mathbb{B}}_{p,\beta}^{n}$ is denoted by

${\mathbb{S}}_{p,\beta}^{n-1}:=\partial{\mathbb{B}}_{p,\beta}^{n}=\left\\{A\in\mathscr{H}_{n}(\mathbb{F}_{\beta}):\sum_{j=1}^{n}|\lambda_{j}(A)|^{p}=1\right\\}$

with the same convention if $p=\infty$. The space
$\mathscr{H}_{n}(\mathbb{F}_{\beta})$ admits a natural scalar product $\langle
A,B\rangle=\operatorname{Re}\operatorname{Tr}(AB^{*})$ so that it becomes a
Euclidean space. The corresponding Riemannian volume on
$\mathscr{H}_{n}(\mathbb{F}_{\beta})$ is denoted by $\text{vol}_{\beta,n}$ and
this measure coincides with the suitably normalized $({\beta n(n-1)\over
2}+n)$-dimensional Hausdorff measure on $\mathscr{H}_{n}(\mathbb{F}_{\beta})$
as follows directly from the area-coarea formula. One can therefore define the
uniform distribution on ${\mathbb{B}}_{p,\beta}^{n}$. The cone probability
measure on ${\mathbb{S}}_{p,\beta}^{n-1}$ is defined as follows: the cone
measure of a Borel set $K\subseteq{\mathbb{S}}_{p,\beta}^{n-1}$ is

$\frac{\text{vol}_{\beta,n}(\cup_{\lambda\in[0,1]}\lambda
K)}{\text{vol}_{\beta,n}({\mathbb{B}}_{p,\beta}^{n})}.$

After having discussed the self-adjoint case, we now turn to the non self-
adjoint one, where the eigenvalues are replaced by the singular values. For an
$(n\times n)$-matrix $A\in\mathrm{Mat}_{n}(\mathbb{F}_{\beta})$ with entries
from the skew field $\mathbb{F}_{\beta}$ with $\beta\in\\{1,2,4\\}$, we denote
by $s_{1}(A),\ldots,s_{n}(A)$ the singular values of $A$. If $\beta=1$ or
$\beta=2$ these are the eigenvalues of $\sqrt{AA^{*}}$, while in the
Hamiltonian case $\beta=4$ we refer to [6, Corollary E.13] for a formal
definition. For $0<p\leq\infty$ the Schatten $p$-ball is defined as

${\mathbb{S}}{\mathbb{B}}_{p}^{n}:=\Big{\\{}A\in\mathrm{Mat}_{n}(\mathbb{F}_{\beta}):\sum_{j=1}^{n}|s_{j}(A)|^{p}\leq
1\Big{\\}}$

with the convention that the sum is replaced by
$\max\\{|s_{j}|:j=1,\ldots,n\\}$ in the case that $p=\infty$. As in the self-
adjoint case, $\mathrm{Mat}_{n}(\mathbb{F}_{\beta})$ can be supplied with the
structure of a Euclidean space in such a way that the $\beta
n^{2}$-dimensional Hausdorff measure restricted to
${\mathbb{S}}{\mathbb{B}}_{p}^{n}$ is finite and can thus be normalized to a
probability measure. Moreover, one can also define the cone probability
measure on the boundary $\partial{\mathbb{S}}{\mathbb{B}}_{p}^{n}$ of
${\mathbb{S}}{\mathbb{B}}_{p}^{n}$.

###  4.2 A non-commutative Schechtman–Zinn probabilistic representation

Let us start with the Schechtman–Zinn type probabilistic representations for
random matrices chosen uniformly at random from ${\mathbb{B}}_{p,\beta}^{n}$
and ${\mathbb{S}}{\mathbb{B}}_{p}^{n}$ for $0<p<\infty$, which are of
independent interest. The basic fact the authors of [66] rely on is indeed a
distributional representation of the empirical measure of a random matrix
$Z_{n}$ chosen from one of the unit balls. We elaborate in slightly more
detail for the case of ${\mathbb{B}}_{p,\beta}^{n}$, before we present the two
results forming the respective backbone.

So assume for each $n\in\mathbb{N}$ we are given a random matrix $Z_{n}$
uniformly distributed in the ball ${\mathbb{B}}_{p,\beta}^{n}$,
$\beta\in\\{1,2,4\\}$. Then one has a probabilistic representation of the
empirical measure of $Z_{n}$ given as

$\mu_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{n^{1/p}\lambda_{i}(Z_{n})}\stackrel{{\scriptstyle
d}}{{=}}\frac{1}{n}\sum_{i=1}^{n}\delta_{n^{1/p}U^{1/\ell}{X_{i,n}\over\|X_{n}\|_{p}}},$

where $\stackrel{{\scriptstyle d}}{{=}}$ denotes equality in distribution and

  * (a)

$X_{n}=(X_{1,n},\dots,X_{n,n})$ is a random vector on $\mathbb{R}^{n}$ with
joint Lebesgue density of the form

$\mathbb{R}^{n}\ni(x_{1},\ldots,x_{n})\mapsto\frac{1}{C_{n,\beta,p}}e^{-n\sum\limits_{i=1}^{n}|x_{i}|^{p}}\prod_{1\leq
i<j\leq n}|x_{i}-x_{j}|^{\beta},$ (13)

with a suitable normalization constant $C_{n,\beta,p}\in(0,\infty)$ depending
on $n$, $\beta$, and $p$,

  * (b)

$U$ is a random variable with uniform distribution on $[0,1]$ that is
independent of $X_{n}$,

  * (c)

$\ell:=\ell(n,\beta):={\beta n(n-1)\over 2}+n$ is the (real) dimension of
$\mathscr{H}_{n}(\mathbb{F}_{\beta})$. 222Let us note that there is a typo in
[66, Item (c)], where $\ell={\beta n(n-1)\over 2}+n\beta$, while it should be
${\beta n(n-1)\over 2}+n$ instead.

Similarly, if $Z_{n}$ is distributed according to the cone measure on
${\mathbb{S}}_{p,\beta}^{n-1}$, then

$\mu_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{n^{1/p}\lambda_{i}(Z_{n})}\stackrel{{\scriptstyle
d}}{{=}}\frac{1}{n}\sum_{i=1}^{n}\delta_{n^{1/p}{X_{i,n}\over\|X_{n}\|_{p}}}.$

We now present the main probabilistic representations used to derive the
level–2 LDPs in Schatten classes. The first result is essentially [65,
Corollary 4.3].

######  Proposition 4.1.

Let $0<p<\infty$ and $Z_{n}$ be uniformly distributed in
${\mathbb{B}}_{p,\beta}^{n}$, $\beta\in\\{1,2,4\\}$. Consider a uniform random
permutation $\pi$ of $\\{1,\ldots,n\\}$, which is independent from $Z_{n}$.
Then

$\big{(}\lambda_{\pi(1)}(Z_{n}),\ldots,\lambda_{\pi(n)}(Z_{n})\big{)}\stackrel{{\scriptstyle
d}}{{=}}U^{1\over n+m}{X_{n}\over\|X_{n}\|_{p}}\qquad\text{with}\qquad
m:=m(n,\beta):={\beta n(n-1)\over 2},$

where $U$ is uniformly distributed on $[0,1]$, and, independently of $U$, the
vector $X_{n}\in\mathbb{R}^{n}$ has joint density proportional to

$\mathbb{R}^{n}\ni(x_{1},\ldots,x_{n})\mapsto
e^{-\sum\limits_{i=1}^{n}|x_{i}|^{p}}\prod_{1\leq i<j\leq
n}|x_{i}-x_{j}|^{\beta},$

where the proportionality constant only depends on $\beta$, $p$, and $n$.
Also, if $Z_{n}\in{\mathbb{S}}_{p,\beta}^{n-1}$ is distributed according to
the cone probability measure, one has that

$\big{(}\lambda_{\pi(1)}(Z_{n}),\ldots,\lambda_{\pi(n)}(Z_{n})\big{)}\stackrel{{\scriptstyle
d}}{{=}}{X_{n}\over\|X_{n}\|_{p}}.$

To present the non self-adjoint variant, for $\beta\in\\{1,2,4\\}$ let us
define the function

$\mathbb{R}_{+}^{n}\to\mathbb{R},\quad(x_{1},\ldots,x_{n})\mapsto
h_{\beta}(x_{1},\ldots,x_{n}):=\prod_{1\leq i<j\leq
n}|x_{i}-x_{j}|^{\beta}\,\prod_{i=1}^{n}x_{i}^{{\beta\over 2}-1},$

which is homogeneous of degree

${\beta n(n-1)\over 2}+n\Big{(}{\beta\over 2}-1\Big{)}.$

The following are now the Schechtman–Zinn-type distributional representations
for Schatten $p$-classes and essentially [66, Proposition 8.2]; note that
instead of $p$ the value $p/2$ occurs at many places since one considers the
squares of the singular values and not the singular values themself.

######  Proposition 4.2.

Let $0<p<\infty$ and $\beta\in\\{1,2,4\\}$. For each $n\in\mathbb{N}$ let
$Z_{n}\in{\mathbb{S}}{\mathbb{B}}_{p}^{n}$ be uniformly distributed and,
independently, let $\pi$ be a uniform random permutation of
$\\{1,\ldots,n\\}$. Then

$(s_{\pi(1)}^{2}(Z_{n}),\ldots,s_{\pi(n)}^{2}(Z_{n}))\stackrel{{\scriptstyle
d}}{{=}}U^{1\over n+m}{X_{n}\over\|X_{n}\|_{p/2}}\qquad\text{with}\qquad
m:=m(n,\beta):={\beta n(n-1)\over 2}+n\Big{(}{\beta\over 2}-1\Big{)},$

where $U$ is uniformly distributed on $[0,1]$, and, independently of $U$, the
random vector $X_{n}\in\mathbb{R}_{+}^{n}$ has joint density proportional to

$\displaystyle\mathbb{R}_{+}^{n}\ni(x_{1},\ldots,x_{n})\mapsto
e^{-n\sum_{i=1}^{n}|x_{i}|^{p/2}}\,h_{\beta}(x_{1},\ldots,x_{n}),$ (14)

where the proportionality constant only depends on $\beta$, $p$, and $n$.
Also, if $Z_{n}\in\partial{\mathbb{S}}{\mathbb{B}}_{p}^{n}$ is distributed
according to the cone probability measure, one has that

$(s_{\pi(1)}^{2}(Z_{n}),\ldots,s_{\pi(n)}^{2}(Z_{n}))\stackrel{{\scriptstyle
d}}{{=}}{X_{n}\over\|X_{n}\|_{p/2}}.$

###  4.3 Sanov-type LDPs for random matrices in Schatten unit balls

In [66] the authors study principles of large deviations in the non-
commutative framework of self-adjoint and non self-adjoint Schatten
$p$-classes. Let us note that an LDP for the law of the spectral measure of a
Gaussian Wigner matrix had already been obtained by G. Ben Arous and A.
Guionnet [18, Theorem 1.1 and Theorem 1.3] and a more general large deviations
theorem for random measures (including the case of the empirical eigenvalue
distribution of Wishart matrices) had been obtained by F. Hiai and D. Petz in
[50, Theorem 1] (see also [52]), which followed their preceding ideas from
[51], where the empirical eigenvalue distribution of suitably distributed
random unitary matrices was shown to satisfies an LDP as the matrix size goes
to infinity. In the same spirit, Z. Kabluchko, J. Prochno, and C. Thäle
obtained Sanov-type LDPs for the spectral measure of $n^{1/p}$ multiples of
random matrices chosen uniformly (or with respect to the cone measure on the
boundary) from the unit balls of self-adjoint and non self-adjoint Schatten
$p$-classes where $0<p\leq\infty$. While the proofs roughly follow a classical
strategy in large deviations theory (see, e.g., [6, 18, 50, 51]), in the
Schatten class case one needs to control simultaneously the deviations of the
empirical measures and their $p$-th moments towards arbitrary small balls in
the product topology of the weak topology on the space of probability measures
and the standard topology on $\mathbb{R}$ in the self-adjoint or
$\mathbb{R}_{+}$ in the non self-adjoint set-up, respectively, and then prove
exponential tightness. A key element is the probabilistic representation for
random points in the unit balls of classical matrix ensembles which we
presented in the previous subsection (see (4.2) and (4.2)) and a non self-
adjoint counterpart (see Proposition 4.2). As we shall see, the good rate
function governing the LDPs is essentially given by the logarithmic energy
(which is remarkably the same as the negative of Voiculescu’s free entropy
introduced in [115]) and, which is quite interesting, a perturbation by a
constant, which is connected to the Ullman distribution. As a consequence of
their LDPs, the authors obtained a law of large numbers and showed that the
spectral measure converges weakly almost surely to the Ullman distribution.

####  4.3.1 LDPs in the self-adjoint case

The following theorem provides a Sanov-type LDP in the self-adjoint setting
and is [66, Theorem 1.1].

######  Theorem G.

Let $0<p<\infty$ and $\beta\in\\{1,2,4\\}$. For every $n\in\mathbb{N}$, let
$Z_{n}$ be a random matrix chosen according to the uniform distribution on
${\mathbb{B}}_{p,\beta}^{n}$ or the cone measure on its boundary
${\mathbb{S}}_{p,\beta}^{n-1}$. Then the sequence of random probability
measures

$\mu_{n}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{n^{1/p}\lambda_{i}(Z_{n})},\qquad
n\in\mathbb{N},$

satisfies an LDP on $\mathcal{M}_{1}(\mathbb{R})$ with speed $n^{2}$ and good
rate function $\mathscr{I}:\mathcal{M}_{1}(\mathbb{R})\to[0,\infty]$ defined
by

$\mathscr{I}(\mu):=\begin{cases}-\frac{\beta}{2}\int_{\mathbb{R}}\int_{\mathbb{R}}\log|x-y|\,\mu(\textup{d}x)\,\mu(\textup{d}y)+\frac{\beta}{2p}\log\left(\frac{\sqrt{\pi}p\Gamma(\frac{p}{2})}{2^{p}\sqrt{e}\Gamma(\frac{p+1}{2})}\right)&\,:\int_{\mathbb{R}}|x|^{p}\mu(\textup{d}x)\leq
1\\\ \infty&\,:\int_{\mathbb{R}}|x|^{p}\mu(\textup{d}x)>1\,.\end{cases}$

######  Remark 4.3.

(i) The distribution of the eigenvalues of a point chosen uniformly at random
in ${\mathbb{B}}_{p,\beta}^{n}$ can be related to the $2$-dimensional Coulomb
gas of $n$ particles at inverse temperature $\beta>0$ in an external potential
$V:t\mapsto|t|^{p}$ acting on each particle.
(ii) As discussed in [66], the additive constant to the logarithmic energy in
the rate function is closely linked to the limit of the free energy whose
precise value follows from results of potential theory.

The next theorem deals with the case $p=\infty$ and corresponds to [66,
Theorem 1.3].

######  Theorem H.

Fix $\beta\in\\{1,2,4\\}$. For every $n\in\mathbb{N}$, let $Z_{n}$ be a random
matrix chosen according to the uniform distribution on
${\mathbb{B}}_{\infty,\beta}^{n}$ or the cone measure on its boundary
${\mathbb{S}}_{\infty,\beta}^{n-1}$. Then the sequence of random probability
measures

$\mu_{n}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{\lambda_{i}(Z_{n})},\qquad
n\in\mathbb{N},$

satisfies an LDP on $\mathcal{M}_{1}(\mathbb{R})$ with speed $n^{2}$ and good
rate function $\mathscr{I}:\mathcal{M}_{1}(\mathbb{R})\to[0,\infty]$ defined
by

$\mathscr{I}(\mu)=\begin{cases}-\frac{\beta}{2}\int_{\mathbb{R}}\int_{\mathbb{R}}\log|x-y|\,\mu(\textup{d}x)\,\mu(\textup{d}y)-\frac{\beta}{2}\log
2&\,:\mu([-1,1])=1\\\ \infty&\,:\mu([-1,1])<1\,.\end{cases}$

As a corollary, we can derive a law of large numbers for $\mu_{n}$ and show
that (weakly almost surely) the sequence of empirical measures converges to a
non-random limiting distribution given by the Ullman measure (for $p<\infty$)
or arcsine measure (for $p=\infty$). We denote the weak convergence of
probability measures by ${\overset{w}{\longrightarrow}}$.

######  Corollary 4.4.

Fix $0<p<\infty$ and $\beta\in\\{1,2,4\\}$. Let $\mu_{\infty}^{(p)}$ be the
probability measure on the interval $[-b_{p},b_{p}]$ with the Ullman density
$x\mapsto h_{p}(x/b_{p})/b_{p}$, where