# Large deviations for uniform projections
of $p$-radial distributions on $\ell_{p}^{n}$-balls

Tom Kaufmann111Ruhr University Bochum, Germany. Email<EMAIL_ADDRESS>Holger Sambale222Ruhr University Bochum, Germany. Email:
<EMAIL_ADDRESS>and Christoph Thäle333Ruhr University Bochum, Germany.
Email<EMAIL_ADDRESS>
###### Abstract

We consider products of uniform random variables from the Stiefel manifold of
orthonormal $k$-frames in $\mathbb{R}^{n}$, $k\leq n$, and random vectors from
the $n$-dimensional $\ell_{p}^{n}$-ball $\mathbb{B}_{p}^{n}$ with certain
$p$-radial distributions, $p\in[1,\infty)$. The distribution of this product
geometrically corresponds to the projection of the $p$-radial distribution on
$\mathbb{B}^{n}_{p}$ onto a random $k$-dimensional subspace. We derive large
deviation principles (LDPs) on the space of probability measures on
$\mathbb{R}^{k}$ for sequences of such projections.
Keywords. Large deviation principle, $\ell_{p}^{n}$-ball, random projection,
Stiefel manifold
MSC: 52A23, 60F10,

##  1 Introduction

The study of high-dimensional convexity goes back to studying infinite-
dimensional normed spaces via their local structures, such as their unit
balls, but has since become of substantial interest in its own right. The
concentration phenomena exhibited by convex objects in high dimensions,
analyzed in the language of probability, are of great use in applications such
as compressed sensing, information theory and approximation theory (see [3, 5,
7, 8, 18]). Analogues of many well known limit results from probability have
been found in high-dimensional convexity, such as the central limit theorem
(CLT) of Klartag [16, 17], and recently, starting with the work of Gantert,
Kim, and Ramanan [6], large deviations began to be considered with increasing
interest as well. For the sake of brevity, we refer the reader to the classic
literature on large deviations theory for more details on basic definitions
and results [4].

Especially the $n$-dimensional $\ell_{p}^{n}$-ball
$\mathbb{B}_{p}^{n}:=\\{x\in\mathbb{R}^{n}:\|x\|_{p}\leq 1\\}$, for
$p\in[1,\infty)$ and $\lVert
x\rVert_{p}:=\big{(}\sum_{i=1}^{n}|x_{i}|^{p}\big{)}^{1/p}$, has been
extensively considered in this regard (see, e.g., [1, 12, 11, 13, 10]) due to
both its relevance in geometry and functional analysis, and its accessibility
via useful probabilistic representation results, which will be outlined below.
For an overview of classical and more recent results about
$\ell_{p}^{n}$-balls, we refer to the survey article [19].
The setting of the present work is a generalization of the one initiated by
Kabluchko and Prochno [10] and can be described as follows. For $k\leq n$, the
Stiefel manifold $\mathbb{V}_{n,k}$ is the set of all orthonormal $k$-frames
in $\mathbb{R}^{n}$, i. e., the set of all $k$-tuples of orthonormal vectors
$v_{1},\ldots,v_{k}$ in $\mathbb{R}^{n}$. Arranging these vectors into a
$k\times n$ matrix $V$ with rows $v_{1}^{T},\ldots,v_{k}^{T}$, we have the
identification

$\mathbb{V}_{n,k}=\\{V\in\mathbb{R}^{k\times n}\colon VV^{T}=I_{k}\\},$

where $I_{k}$ denotes the $k\times k$ identity matrix. We equip
$\mathbb{V}_{n,k}$ with the uniform distribution (i. e., the invariant Haar
probability measure) $\mu_{n,k}$, writing $V_{n,k}$ for the corresponding
random variable. Recall that $V_{n,k}$ is characterized by the following
invariance property: for any orthogonal matrices $O\in\mathbb{R}^{k\times k}$
and $O^{\prime}\in\mathbb{R}^{n\times n}$, $OV_{n,k}O^{\prime}$ has the same
distribution as $V_{n,k}$.

In particular, for random vectors $X^{(n)}$ taking values in $\mathbb{R}^{n}$,
we may regard $V\in\mathbb{V}_{n,k}$ as a linear map
$V\colon\mathbb{R}^{n}\to\mathbb{R}^{k}$ and study the distribution of the
vectors $VX^{(n)}\in\mathbb{R}^{k}$, which we denote by

$\mu_{VX^{(n)}}(A):=\mathbb{P}(VX^{(n)}\in A)$

for any Borel set $A\subseteq\mathbb{R}^{k}$. In addition, we may also choose
$V_{n,k}\in\mathbb{V}_{n,k}$ at random according to the uniform distribution
$\mu_{n,k}$ on $\mathbb{V}_{n,k}$. In this case, the distribution of
$V_{n,k}X^{(n)}$, which we denote by

$\mu_{V_{n,k}X^{(n)}}(A):=\mathbb{P}(V_{n,k}X^{(n)}\in A),$ (1.1)

is a random probability measure on $\mathbb{R}^{k}$, that is a random variable
taking values in the space $\mathcal{M}_{1}(\mathbb{R}^{k})$ of probability
measures on $\mathbb{R}^{k}$, which we equip with the topology of weak
convergence. This can geometrically be interpreted as the projection of the
distribution of the random vector $X^{(n)}$ onto a uniform random
$k$-dimensional subspace.

We are interested in large deviation principles (LDPs) for the random
probability measures $\mu_{V_{n,k}X^{(n)}}$, where
$X^{(n)}\in\mathbb{B}_{p}^{n}$ with a distribution taken from the class of
$p$-radial distributions introduced by Barthe, Gudéon, Mendelson and Naor in
[2]. Kabluchko and Prochno [10] gave very general LDPs for random matrices in
the orthogonal group and the Stiefel manifold, and showed an LDP for
$k$-dimensional projections of the special case of the uniform distribution on
$\mathbb{B}_{p}^{n}$ as an application. Based on [10], our work largely
extends the set of projected distributions for which such an LDP is shown from
the uniform distribution to the aforementioned class of $p$-radial
distributions. In particular, we will see that the large deviation behaviour
observed by Kabluchko and Prochno [10] is universal for a large class of
probability measures on $\mathbb{B}_{p}^{n}$. Moreover, we shall describe
geometrically motivated distributions on $\mathbb{B}_{p}^{n}$ for which the
LDP needs a suitable modification we also provide. We should also delineate
the present work from the results shown by Kim and Ramanan in [15, Theorems
2.4 & 2.6], who have shown, among other results, LDPs for uniform random
projections of uniform random vectors in $\mathbb{B}^{n}_{p}$ onto
$k$-dimensional subspaces. By the same arguments as put forth in [10], we note
that while the settings are quite similar, the key difference is in the object
of study, which in [15] is the projection point itself, hence yielding an LDP
on $\mathbb{R}^{k}$, whereas in both [10] and this work it is the projected
distribution on $\mathbb{R}^{k}$, thus the main result yields an LDP on the
space $\mathcal{M}_{1}(\mathbb{R}^{k})$ of probability measures on
$\mathbb{R}^{k}$.

In the next section we briefly list the notation and background material we
will need to formulate our theorems, which in turn are presented in Section 3.
Section 4 will then contain their respective proofs.

##  2 Preliminaries

Let us first define the objects and distributions needed for the main results.
We write $\mathcal{B}(\mathbb{R}^{n})$ for the Borel $\sigma$-algebra,
$\langle\,\cdot\,,\,\cdot\,\rangle_{2}$ for the Euclidean scalar product, and
$\mathrm{vol}_{n}(\cdot)$ for the Lebesgue measure on $\mathbb{R}^{n}$. As
already mentioned in the introduction, for $p\in[1,\infty)$, $n\in\mathbb{N}$,
and $x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$ we denote by
$\|x\|_{p}:=\big{(}\sum\limits_{i=1}^{n}|x_{i}|^{p}\big{)}^{1/p}$ the
$\ell_{p}^{n}$-norm of $x$ and by
$\mathbb{B}_{p}^{n}:=\\{x\in\mathbb{R}^{n}:\|x\|_{p}\leq 1\\}$ and
$\SS_{p}^{n-1}:=\\{x\in\mathbb{R}^{n}:\|x\|_{p}=1\\}$ unit $\ell_{p}^{n}$-ball
and unit $\ell_{p}^{n}$-sphere, respectively. We define the uniform
distribution on $\mathbb{B}^{n}_{p}$ and the cone probability measure on
$\SS^{n-1}_{p}$ as

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

Following [2], for a sequence $(\mathbf{W}_{n})_{n\in\mathbb{N}}$ of Borel
probability measures $\mathbf{W}_{n}$ on $[0,\infty)$ we define the sequence
of distributions

$\mathbf{P}_{n,p,\mathbf{W}_{n}}:=\mathbf{W}_{n}(\\{0\\})\mathbf{C}_{n,p}+\Psi_{n}\mathbf{U}_{n,p}$
(2.1)

on $\mathbb{B}^{n}_{p}$, where $\Psi_{n}(x)=\psi(\|x\|_{p})$,
$x\in\mathbb{B}_{p}^{n}$, is the $p$-radial density given by

$\displaystyle\psi_{n}(s)=\displaystyle{1\over p^{n/p}\Gamma\big{(}{n\over
p}+1\big{)}}{1\over(1-s^{p})^{{n\over p}+1}}\bigg{[}\int_{(0,\infty)}w^{n\over
p}\,e^{-\frac{1}{p}\big{(}{s^{p}\over
1-s^{p}}\big{)}w}\,\mathbf{W}_{n}(\textup{d}w)\bigg{]},\qquad 0\leq s\leq 1.$

One can think of $\mathbf{W}_{n}$ as indicating how probability mass is
distributed $p$-radially within $\mathbb{B}_{p}^{n}$. The motivation behind
this class of distributions is twofold. First, they encompass many relevant
distributions on $\mathbb{B}_{p}^{n}$. For instance, choosing
$\mathbf{W}_{n}\equiv\delta_{0}$ to be the Dirac measure at $0$, we have that
$\mathbf{P}_{n,p,\mathbf{W}_{n}}\equiv\mathbf{C}_{n,p}$ and for
$\mathbf{W}_{n}\equiv\mathrm{Exp}(1)$, we have that
$\mathbf{P}_{n,p,\mathbf{W}_{n}}\equiv\mathbf{U}_{n,p}$. For $m\in\mathbb{N}$
choosing $\mathbf{W}_{n}=\gamma(\frac{m}{p},\frac{1}{p})$, i.e., a gamma
distribution with shape $m/p$ and rate $1/p$, it can be shown that
$\mathbf{P}_{n,p,\mathbf{W}_{n}}$ then corresponds to the projection of
$\mathbf{C}_{n+m,p}$ onto its first $n$ coordinates. An analogue
correspondence is given for $\mathbf{W}_{n}=\gamma(1+\frac{m}{p},\frac{1}{p})$
and the projection of $\mathbf{U}_{n+m,p}$ onto its first $n$ coordinates, see
[2]. The second reason we consider the class of distributions
$\mathbf{P}_{n,p,\mathbf{W}_{n}}$ specifically is a useful probabilistic
representation result. For $p\in[1,\infty)$ we say a real-valued random
variable $X$ has a $p$-generalized Gaussian distribution, denoted as
$X\sim\mathbf{N}_{p}$, if its distribution has Lebesgue density

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

For $X\sim\mathbf{N}_{p}$ and $r>0$ the $r$-th moment of $X$ is given by

$\mathbb{E}\left[X^{r}\right]=\Gamma\left(\frac{1+r}{p}\right){\Gamma\left(\frac{1}{p}\right)}^{-1}.$
(2.2)

In particular, $\mathbb{E}[X^{r}]<\infty$ for all $r>0$. Using this
$p$-generalized Gaussian distribution, the following results from [2, Theorem
3] gives a way to represent a random vector
$X^{(n)}\sim\mathbf{P}_{n,p,\mathbf{W}_{n}}$ in $\mathbb{B}_{p}^{n}$ as a
vector with i.i.d. $p$-generalized Gaussian coordinates, normalized by its
norm and via $\mathbf{W}_{n}$.

######  Proposition 2.1.

Let $n\in\mathbb{N}$ and $p\in(0,\infty)$. Let $Z^{(n)}=(Z_{1},\ldots,Z_{n})$
be a random vector, where $Z_{1},\ldots,Z_{n}$ are i.i.d. with
$Z_{i}\sim\mathbf{N}_{p}$, and $W_{n}$ a random variable with distribution
$\mathbf{W}_{n}$ on $[0,\infty)$, independent of $Z^{(n)}$. Then the random
vector

$Z^{(n)}\over(\|Z^{(n)}\|_{p}^{p}+W_{n})^{1/p}$

has distribution $\mathbf{P}_{n,p,\mathbf{W}_{n}}$ as in (2.1).

The final result presented in this section is one of the main results from
[10], namely Theorem D therein. It provides an LDP for random projections of
product measures, which we will use in conjunction with the representation
from Proposition 2.1 to prove our theorems. In what follows we shall write
$\mathcal{D}(X)$ for the distribution of a random variable $X$. Moreover, let

$\mathcal{R}_{2}^{k\times\infty}:=\\{A=(A_{ij})_{i,j=1}^{k,\infty}\colon(A_{ij})_{j\in\mathbb{N}}\in\ell_{2},i=1,\ldots,k\\}$

be the set of all matrices $A\in\mathbb{R}^{k\times\infty}$ with square-
summable rows. For $A\in\mathbb{R}^{k\times\infty}$ we denote by $\lVert
AA^{T}\rVert_{\mathrm{op}}$ the operator norm of the matrix
$AA^{T}\in\mathbb{R}^{k\times k}$, where the condition
$A\in\mathcal{R}_{2}^{k\times\infty}$ guarantees that $AA^{T}$ is well-
defined.

######  Proposition 2.2.

Fix $k\in\mathbb{N}$. For each $n\in\mathbb{N}$ let
$Z^{(n)}=(Z_{1},\ldots,Z_{n})$ be an $n$-dimensional random vector, where
$Z_{1},Z_{2},\ldots$ are i.i.d. non-Gaussian random variables with symmetric
distribution and finite moments of all orders. Let
$\sigma^{2}:=\mathbb{E}[Z_{1}^{2}]>0$ be the variance of $Z_{1}$. Then, the
sequence of random probability measures $\mu_{V_{n,k}Z^{(n)}}$, $n\geq k$, as
in (1.1) satisfies an LDP on $\mathcal{M}_{1}(\mathbb{R}^{k})$ with speed $n$
and good rate function $I\colon\mathcal{M}_{1}(\mathbb{R}^{k})\to[0,\infty]$
given by

$I(\nu)=-\frac{1}{2}\log\mathrm{det}(I_{k}-AA^{T})$

if $\nu$ admits a representation of the form

$\nu=\mathcal{D}\Big{(}\sum_{j=1}^{\infty}A_{\bullet,j}Z_{j}+\sigma(I_{k}-AA^{T})^{1/2}N_{k}\Big{)}$

for some matrix $A\in\mathcal{R}_{2}^{k\times\infty}$ with columns
$A_{\bullet,1},A_{\bullet,2},\ldots$ such that $\lVert
AA^{T}\rVert_{\mathrm{op}}<1$, where $N_{k}$ is a $k$-dimensional standard
Gaussian random vector independent of $Z_{1},Z_{2},\ldots$. If $\nu$ does not
admit a representation of this form, $I(\nu)=\infty$.

Note that the specific distribution of the $Z_{i}$ has a rather subtle
influence on the rate function of the LDP via the matrix $A$ used in the
representation of a given measure $\nu\in\mathcal{M}_{1}(\mathbb{R}^{k})$. As
a side remark, note that in [10, Theorem D], the case of $\sigma^{2}=0$
actually has to be excluded. We have amended the result accordingly.

##  3 Main Results

We are now in the position to present the first of our main results for the
projections of $p$-radial distributions $\mathbf{P}_{n,p,\mathbf{W}_{n}}$ on
$\ell_{p}^{n}$-balls. In what follows we shall write
$\overset{\mathcal{D}}{=}$ for equality in distribution.

######  Theorem 3.1.

Fix $p\in[1,\infty)$, $p\neq 2$, and $k\in\mathbb{N}$. Moreover, let
$(\mathbf{W}_{n})_{n\in\mathbb{N}}$ be a sequence of Borel probability
measures on $[0,\infty)$ and $(W_{n})_{n\in\mathbb{N}}$ a sequence of random
variables with $W_{n}\sim\mathbf{W}_{n}$, such that
$W_{n}/n\to\alpha\in[0,\infty)$ in probability. Finally, let $X^{(n)},Y^{(n)}$
be random vectors in $\mathbb{B}_{p}^{n}$ with
$Y^{(n)}\sim\mathbf{P}_{n,p,\mathbf{W}_{n}}$ and
$X^{(n)}\overset{\mathcal{D}}{=}n^{1/p}\,Y^{(n)}$. Then, the sequence of
random probability measures $\mu_{V_{n,k}X^{(n)}}$, $n\geq k$, as in (1.1)
satisfies an LDP on $\mathcal{M}_{1}(\mathbb{R}^{k})$ with speed $n$ and good
rate function $I\colon\mathcal{M}_{1}(\mathbb{R}^{k})\to[0,\infty]$ given by

$I(\nu)=-\frac{1}{2}\log\mathrm{det}(I_{k}-AA^{T})$

if $\nu$ admits a representation of the form

$\nu=\mathcal{D}\Big{(}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}\sum_{j=1}^{\infty}A_{\bullet,j}Z_{j}+\sigma_{p,\alpha}(I_{k}-AA^{T})^{1/2}N_{k}\Big{)}$

for some matrix $A\in\mathcal{R}_{2}^{k\times\infty}$ with columns
$A_{\bullet,1},A_{\bullet,2},\ldots$ such that $\lVert
AA^{T}\rVert_{\mathrm{op}}<1$, where $Z_{1},Z_{2},\ldots$ are i.i.d. with
$Z_{i}\sim\mathbf{N}_{p}$,

$\sigma_{p,\alpha}^{2}:=\Big{(}\frac{p}{1+\alpha}\Big{)}^{2/p}\frac{\Gamma(3/p)}{\Gamma(1/p)},$

and $N_{k}$ is an independent $k$-dimensional standard Gaussian random vector.
If $\nu$ does not admit a representation of this form, $I(\nu)=\infty$.

As discussed earlier, choosing $\mathbf{W}_{n}\equiv\delta_{0}$ gives
$\mathbf{P}_{n,p,\mathbf{W}_{n}}\equiv\mathbf{C}_{n,p}$ and
$\mathbf{W}_{n}\equiv\mathrm{Exp}(1)$ yields
$\mathbf{P}_{n,p,\mathbf{W}_{n}}\equiv\mathbf{U}_{n,p}$, in both cases it
holds for $W_{n}\sim\mathbf{W}_{n}$ that $W_{n}/n\to 0$ in probability and we
get back [10, Theorem C]. Hence, we can see that both $\mathbf{C}_{n,p}$ and
$\mathbf{U}_{n,p}$ share the same LDP behaviour in high dimensions, which is
in line with similar observations made for other functionals (see, e.g., [1,
12, 14]). Moreover, the result even implies a certain universality of the rate
function, since despite the expected sensitivity of LDPs to the underlying
distributions, the rate function is the same for all sequences
$(\mathbf{W}_{n})_{n\in\mathbb{N}}$ that share the same limiting behaviour.

Given the setting of Theorem 3.1, if we consider the case $W_{n}/n\to\infty$
in probability (formally corresponding to the choice $\alpha=\infty$), by the
representation result in Proposition 2.1 one can see that this corresponds to
each component of $X^{(n)}$ converging to $0$ in probability, that is, we
arrive at a trivial limit. To avoid this, we may choose a different scaling as
carried out in the following theorem.

######  Theorem 3.2.

Fix $p\in[1,\infty)$, $p\neq 2$, and $k\in\mathbb{N}$. Moreover, let
$(\mathbf{W}_{n})_{n\in\mathbb{N}}$ be a sequence of Borel probability
measures on $[0,\infty)$ and $(W_{n})_{n\in\mathbb{N}}$ a sequence of random
variables with $W_{n}\sim\mathbf{W}_{n}$ and
$W_{n}/n^{\kappa}\to\beta\in(0,\infty)$ in probability for some $\kappa>1$,
and assume that the sequence of random variables $(W_{n}/n^{\kappa})^{-2/p}$
is uniformly integrable. Finally, let $X^{(n)},Y^{(n)}$ be random vectors in
$\mathbb{B}_{p}^{n}$ with $Y^{(n)}\sim\mathbf{P}_{n,p,\mathbf{W}_{n}}$ and
$X^{(n)}\overset{\mathcal{D}}{=}n^{\kappa/p}\,Y^{(n)}$. Then, the sequence of
random probability measures $\mu_{V_{n,k}X^{(n)}}$, $n\geq k$, as in (1.1)
satisfies an LDP on $\mathcal{M}_{1}(\mathbb{R}^{k})$ with speed $n$ and good
rate function $I\colon\mathcal{M}_{1}(\mathbb{R}^{k})\to[0,\infty]$ given by

$I(\nu)=-\frac{1}{2}\log\mathrm{det}(I_{k}-AA^{T})$

if $\nu$ admits a representation of the form

$\nu=\mathcal{D}\Big{(}\Big{(}\frac{1}{\beta}\Big{)}^{1/p}\sum_{j=1}^{\infty}A_{\bullet,j}Z_{j}+\sigma_{p,\beta}(I_{k}-AA^{T})^{1/2}N_{k}\Big{)}$

for some matrix $A\in\mathcal{R}_{2}^{k\times\infty}$ with columns
$A_{\bullet,1},A_{\bullet,2},\ldots$ such that $\lVert
AA^{T}\rVert_{\mathrm{op}}<1$, where $Z_{1},Z_{2},\ldots$ are i.i.d.
$p$-generalized Gaussian random variables,

$\sigma_{p,\beta}^{2}:=\Big{(}\frac{p}{\beta}\Big{)}^{2/p}\frac{\Gamma(3/p)}{\Gamma(1/p)},$

and $N_{k}$ is an independent $k$-dimensional standard Gaussian random vector.
If $\nu$ does not admit a representation of this form, $I(\nu)=\infty$.

Note that a helpful sufficient condition for the uniform integrability of
$(W_{n}/n)^{-2/p}$ is given by

$\sup_{n\in\mathbb{N}}\,\mathbb{E}\left[\Big{(}\frac{n^{\kappa}}{W_{n}}\Big{)}^{4/p}\right]\leq
C$ (3.1)

for some absolute constant $C>0$. In particular, it can be applied to verify
the uniform integrability for certain gamma distributions.

######  Lemma 3.3.

Suppose for each $n\in\mathbb{N}$ that $W_{n}$ follows a gamma distribution
with shape $a_{n}$ and rate $b>0$, where $(a_{n})_{n\in\mathbb{N}}$ is a
positive increasing sequence and $b>0$. We assume that $a_{n}$ satisfies
$\inf_{n}a_{n}=:m>4/p$ and $\lim_{n\to\infty}{a_{n}\over
n^{\kappa}}=\lambda\in(0,\infty)$ for some $\kappa\in(0,\infty)$. Then

$\sup_{n\in\mathbb{N}}\,\mathbb{E}\left[\Big{(}\frac{n^{\kappa}}{W_{n}}\Big{)}^{4/p}\right]\leq
b^{4/p}M_{p}(\lambda m)^{-4/p}<\infty,$ (3.2)

where $M_{p}^{-1}=\prod_{i=0}^{4}(1-{4\over p(m+i)})$.

The proof of this lemma is postponed to the end of this paper. As a concrete
and geometrically motivated example we consider the distribution on
$\mathbb{B}_{p}^{n}$ arising as the projection to the first $n$ coordinates of
the cone probability measure $\mathbf{C}_{n+m_{n},p}$ on
$\mathbb{B}_{p}^{n+m_{n}}$, where $m_{n}$ is an increasing sequence satisfying
$\inf_{n}m_{n}=m>4$ and $\lim_{n\to\infty}{m_{n}\over n^{\kappa}}=\lambda$ for
some $\kappa\geq 1$ and $\lambda\in(0,\infty)$. As discussed above, this case
corresponds to $\mathbf{P}_{n,p,\mathbf{W}_{n}}$ with
$\mathbf{W}_{n}=\gamma({m_{n}\over p},{1\over p})$ and fits the assumptions of
Lemma 3.3. The same holds for the projection of the uniform distribution
$\mathbf{U}_{n+m_{n},p}$ corresponding to $\mathbf{P}_{n,p,\mathbf{W}_{n}}$
with $\mathbf{W}_{n}=\gamma(1+{m_{n}\over p},{1\over p})$. In particular,
Theorem 3.2 applies to these situations.

##  4 Proofs

This section shall prove Theorem 3.1 and Theorem 3.2. The proofs will follow
in the footsteps of the proof of [10, Theorem C], adapting and generalizing
the arguments where necessary. We start off by formulating some probabilistic
representations of the target quantities and show some auxiliary results for
the proof.

Assume the set-up of Theorem 3.1 and for a fixed Stiefel matrix
$V\in\mathbb{V}_{n,k}$ denote by $V_{\bullet,j}$, $j=1,\ldots,n$ its columns.
Then, by (1.1) and the representation results from Proposition 2.1 it follows
that for any Borel set $A\in\mathcal{B}(\mathbb{R}^{k})$,

$\mu_{VX^{(n)}}(A)=\mathbb{P}(VX^{(n)}\in
A)=\mathbb{P}\Big{(}\sum_{j=1}^{n}n^{1/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\in A\Big{)},$ (4.1)

where $Z^{(n)}=(Z_{1},\ldots,Z_{n})$ with $Z_{j}\sim\mathbf{N}_{p}$ i.i.d. and
$W_{n}\sim\mathbf{W}_{n}$ independent of $Z^{(n)}$. Moreover, let

$\tilde{\mu}_{VX^{(n)}}(A):=\mathbb{P}\Big{(}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}\in
A\Big{)},$ (4.2)

again with i.i.d. $Z_{j}\sim\mathbf{N}_{p}$. We shall see that we can confine
our analysis to $\tilde{\mu}_{VX^{(n)}}$ instead of $\mu_{VX^{(n)}}$, since
they are arbitrarily close to each other in $n\in\mathbb{N}$ with respect to
the Lévy-Prokhorov metric. On the space $\mathcal{M}_{1}(\mathbb{R}^{k})$ of
probability measures on $\mathbb{R}^{k}$, the Lévy–Prokhorov metric
$\rho_{\mathrm{LP}}$ is defined by

$\rho_{\mathrm{LP}}(\mu,\nu):=\inf\\{\varepsilon>0\colon\mu(A)\leq\nu(A_{\varepsilon})+\varepsilon\text{
and }\nu(A)\leq\mu(A_{\varepsilon})+\varepsilon\text{ for all
$A\in\mathcal{B}(\mathbb{R}^{k})$}\\},$

where $A_{\varepsilon}$ denotes the $\varepsilon$-neighborhood of
$A\in\mathcal{B}(\mathbb{R}^{k})$, defined as

$A_{\varepsilon}:=\\{x\in\mathbb{R}^{k}\colon\|a-x\|_{2}<\varepsilon\text{ for
some $a\in A$}\\},\qquad\varepsilon>0.$

We shall now prove that for the Lévy–Prokhorov metric $\rho_{\mathrm{LP}}$ on
$\mathcal{M}_{1}(\mathbb{R}^{k})$,
$\rho_{\mathrm{LP}}(\mu_{V},\tilde{\mu}_{V})$ converges to $0$ uniformly over
all $V\in\mathbb{V}_{n,k}$, as $n\to\infty$.

######  Lemma 4.1.

For $p\in[1,\infty)$ and any $n\in\mathbb{N}$ set $X^{(n)}$ as in Theorem 3.1.
Then, for $k\leq n$, we have

$\lim_{n\to\infty}\sup_{V\in\mathbb{V}_{n,k}}\rho_{\mathrm{LP}}(\mu_{VX^{(n)}},\tilde{\mu}_{VX^{(n)}})=0.$

###### Proof.

Let $A\in\mathcal{B}(\mathbb{R}^{k})$, and $\varepsilon>0$. Then,

$\displaystyle\tilde{\mu}_{VX^{(n)}}(A)$
$\displaystyle=\mathbb{P}\Big{(}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}\in
A\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\sum_{j=1}^{n}n^{1/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\in A_{\varepsilon}\Big{)}$
$\displaystyle\quad+\mathbb{P}\Big{(}\Big{\lVert}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}-\sum_{j=1}^{n}n^{1/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\Big{\rVert}_{2}\geq\varepsilon\Big{)}$
$\displaystyle=\mu_{VX^{(n)}}(A_{\varepsilon})+\mathbb{P}\Big{(}\Big{\lVert}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}-\sum_{j=1}^{n}n^{1/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\Big{\rVert}_{2}\geq\varepsilon\Big{)}.$
(4.3)

Let us prove that the second summand on the right-hand side converges to $0$,
as $n\to\infty$. By Markov’s inequality,

$\displaystyle\mathbb{P}\Big{(}\Big{\lVert}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}-\sum_{j=1}^{n}n^{1/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\Big{\rVert}_{2}\geq\varepsilon\Big{)}$
$\displaystyle\hskip
56.9055pt\leq\varepsilon^{-1}\mathbb{E}\Big{\lVert}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}-\sum_{j=1}^{n}n^{1/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\Big{\rVert}_{2},$

and by the Cauchy–Schwarz inequality,

$\displaystyle\mathbb{E}\Big{\lVert}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}-\sum_{j=1}^{n}n^{1/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\Big{\rVert}_{2}$
$\displaystyle\qquad=\mathbb{E}\Big{(}\Big{\lVert}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}\Big{\lVert}_{2}\cdot\Big{|}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}-\frac{n^{1/p}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}\Big{|}\Big{)}$
$\displaystyle\qquad\leq\sqrt{\mathbb{E}\Big{\lVert}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}\Big{\lVert}_{2}^{2}}\cdot\sqrt{\mathbb{E}\Big{|}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}-\frac{n^{1/p}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}\Big{|}^{2}}.$ (4.4)

As $Z_{1},\ldots,Z_{n}$ are i.i.d. with mean zero and
$V_{\bullet,1},\ldots,V_{\bullet,n}$ are orthonormal vectors, the first factor
in (4.4) reads

$\mathbb{E}\Big{\lVert}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}\Big{\lVert}_{2}^{2}=\mathbb{E}\Big{[}\sum_{i,j=1}^{n}Z_{i}Z_{j}\langle
V_{\bullet,i},V_{\bullet,j}\rangle_{2}\Big{]}=\mathbb{E}[Z_{1}^{2}]\sum_{j=1}^{n}\langle
V_{\bullet,i},V_{\bullet,j}\rangle_{2}=k\mathbb{E}[Z_{1}^{2}].$ (4.5)

To address the second factor in (4.4), let us first argue that

$\xi_{n}:=\Big{(}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}-\frac{n^{1/p}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}\Big{)}^{2}\longrightarrow 0$ (4.6)

in probability, as $n\to\infty$. Indeed, by the continuous mapping theorem, it
suffices to show that

$\frac{\lVert Z^{(n)}\rVert_{p}^{p}}{n}+\frac{W_{n}}{n}\longrightarrow
1+\alpha$

in probability. This follows from the fact that as $Z_{1},\ldots,Z_{n}$ are
i.i.d. $p$-generalized Gaussian random variables, we have
$\mathbb{E}|Z_{i}|^{p}=1$, and moreover that by assumption,
$W_{n}/n\longrightarrow\alpha$ in probability. In fact, we even have
$\xi_{n}\longrightarrow 0$ in $L^{1}$. To see this, it suffices to show that
$(\xi_{n})_{n}$ is uniformly integrable, which in combination with (4.6)
yields convergence in $L^{1}$. Clearly, $(\xi_{n})_{n}$ is uniformly
integrable if the sequence

$\Big{(}\frac{n}{\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n}}\Big{)}^{2/p}\leq\Big{(}\frac{n}{\lVert
Z\rVert_{p}^{p}}\Big{)}^{2/p}$

is uniformly integrable, where we have used that $W_{n}\geq 0$. This in turn
follows from the fact that $\|Z^{(n)}\|_{p}^{p}\sim\gamma(n/p,1/p)$ together
with (3.2) for $\mathbf{W}_{n}=\gamma(n/p,1/p)$, $\kappa=1$ and $\lambda=1/p$,
which yields

$\mathbb{E}\Big{(}\frac{n}{\lVert Z^{(n)}\rVert_{p}^{p}}\Big{)}^{4/p}\leq
p^{4/p}M_{p}\in(0,\infty)$

for all $n\in\mathbb{N}$. Hence, $\xi_{n}\longrightarrow 0$ in $L^{1}$, and as
a consequence, the second factor in (4.4) converges to $0$. This implies that
the second summand in (4.3) converges to $0$ uniformly in
$V\in\mathbb{V}_{n,k}$. Altogether, we have proven that for any
$\varepsilon>0$,

$\tilde{\mu}_{VX^{(n)}}(A)\leq\mu_{VX^{(n)}}(A_{\varepsilon})+\varepsilon$

for $n$ sufficiently large. In the same way, we may also prove that

$\mu_{VX^{(n)}}(A)\leq\tilde{\mu}_{VX^{(n)}}(A_{\varepsilon})+\varepsilon$

for $n$ sufficiently large, which finishes the proof. ∎

Finally, let us replace $V\in\mathbb{V}_{n,k}$ by random variables $V_{n,k}$,
i. e. the Stiefel matrix is chosen at random according to the uniform
distribution $\mu_{n,k}$ on $\mathbb{V}_{n,k}$. Based on Lemma 4.1, we may
prove that a weak LDP for the modified sequence $\tilde{\mu}_{V_{n,k}X^{(n)}}$
implies a weak LDP (in the sense of [4, Definition, p.7]) for
$\mu_{V_{n,k}X^{(n)}}$, both respectively defined as in (4.1) and (4.2) with
respect to $V_{n,k}$.

######  Lemma 4.2.

Assume the set-up of Theorem 3.1 and recall the notation (4.1) and (4.2). If
the sequence $\tilde{\mu}_{V_{n,k}X^{(n)}}$ satisfies a weak LDP on
$\mathcal{M}_{1}(\mathbb{R}^{k})$ at speed $n$ and rate function $I$, then the
sequence $\mu_{V_{n,k}X^{(n)}}$ satisfies the same weak LDP.

###### Proof.

It suffices to check the weak LDP on a basis of the topology of
$\mathcal{M}_{1}(\mathbb{R}^{k})$, e. g., the balls

$B_{r}(\nu):=\\{\mu\in\mathcal{M}_{1}(\mathbb{R}^{k})\colon\rho_{\mathrm{LP}}(\mu,\nu)<r\\}$

for any $r\in(0,\infty)$. By Lemma 4.1, for $n$ sufficiently large we have
$\rho_{\mathrm{LP}}(\tilde{\mu}_{V_{n,k}X^{(n)}},\mu_{V_{n,k}X^{(n)}})<r/2$
uniformly over all realizations of $V_{n,k}\in\mathbb{V}_{n,k}$. Therefore, by
the triangle inequality for $\rho_{\mathrm{LP}}$,

$\frac{1}{n}\log\mathbb{P}(\tilde{\mu}_{V_{n,k}X^{(n)}}\in
B_{r/2}(\nu))\leq\frac{1}{n}\log\mathbb{P}(\mu_{V_{n,k}X^{(n)}}\in
B_{r}(\nu))\leq\frac{1}{n}\log\mathbb{P}(\tilde{\mu}_{V_{n,k}X^{(n)}}\in
B_{3r/2}(\nu)),$

and hence,

$\displaystyle\limsup_{n\to\infty}\frac{1}{n}\log\mathbb{P}(\tilde{\mu}_{V_{n,k}X^{(n)}}\in
B_{r/2}(\nu))$
$\displaystyle\leq\liminf_{n\to\infty}\frac{1}{n}\log\mathbb{P}(\mu_{V_{n,k}X^{(n)}}\in
B_{r}(\nu))$
$\displaystyle\leq\limsup_{n\to\infty}\frac{1}{n}\log\mathbb{P}(\mu_{V_{n,k}X^{(n)}}\in
B_{r}(\nu))\leq\liminf_{n\to\infty}\frac{1}{n}\log\mathbb{P}(\tilde{\mu}_{V_{n,k}X^{(n)}}\in
B_{3r/2}(\nu)).$

Thus, by monotonicity in $r$, taking the infimum over $r\in(0,\infty)$, the
LDP for $\tilde{\mu}_{V_{n,k}X^{(n)}}$ yields

$-I(\nu)\leq\inf_{r\in(0,\infty)}\liminf_{n\to\infty}\frac{1}{n}\log\mathbb{P}(\mu_{V_{n,k}X^{(n)}}\\!\in\\!B_{r}(\nu))\leq\inf_{r\in(0,\infty)}\limsup_{n\to\infty}\frac{1}{n}\log\mathbb{P}(\mu_{V_{n,k}X^{(n)}}\\!\in\\!B_{r}(\nu))\leq-I(\nu).$

From here the claim follows from [4, Theorem 4.1.11]. ∎

On a compact space, weak and full LDPs coincide. Here, compactness is provided
by the following lemma.

######  Lemma 4.3.

There is a constant $C\in(0,\infty)$ such that for all $n\geq k$ and all
$V\in\mathbb{V}_{n,k}$,

$\mu_{VX^{(n)}}\in
M_{C}:=\Big{\\{}\mu\in\mathcal{M}_{1}(\mathbb{R}^{k})\colon\int_{\mathbb{R}^{k}}\lVert
x\rVert_{2}\,\mu(\textup{d}x)\leq C\Big{\\}},$

where the set $M_{C}$ is compact for any choice of $C\in(0,\infty)$.

###### Proof.

The compactness of the set $M_{C}$ in the weak topology on
$\mathcal{M}_{1}(\mathbb{R}^{k})$ has been shown in [10, Proof of Lemma 5.3],
so it remains to prove the first assertion. To this end, recalling the
representation of the distribution $\mu_{VX^{(n)}}$ given in (4.1), it
suffices to prove that

$\limsup_{n\to\infty}\sup_{V\in\mathbb{V}_{n,k}}\mathbb{E}\Big{\lVert}\sum_{j=1}^{n}n^{1/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\Big{\rVert}_{2}<\infty,$

for i.i.d. $Z_{j}\sim\mathbf{N}_{p}$ and $W_{n}\sim\mathbf{W}_{n}$ as in
Theorem 3.1. By the triangle inequality it then follows that

$\displaystyle\mathbb{E}\Big{\lVert}\sum_{j=1}^{n}n^{1/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\Big{\rVert}_{2}$
$\displaystyle\leq\mathbb{E}\Big{\lVert}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}-\sum_{j=1}^{n}n^{1/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\Big{\rVert}_{2}+\mathbb{E}\Big{\lVert}\Big{(}\frac{1}{1+\alpha}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}\Big{\rVert}_{2}.$

The first summand on the right hand side converges to $0$ uniformly in
$V_{n,k}\in\mathbb{V}_{n,k}$ as was shown after (4.4). Moreover, by Hölder’s
inequality and (4.5), the second summand is uniformly bounded by
$\sqrt{k\mathbb{E}[Z_{1}^{2}]}/(1+\alpha)^{1/p}$, and thus, the claim follows.
∎

Combining the accumulated auxiliary results, we now have the sufficient tools
to prove Theorem 3.1.

###### Proof of Theorem 3.1.

Apply Proposition 2.2 to the symmetric non-Gaussian random variables
$Z_{j}/(1+\alpha)^{1/p}$, which, by (2.2), have finite moments of all orders
and, in particular, variance

$\sigma_{p,\alpha}^{2}=\Big{(}\frac{p}{1+\alpha}\Big{)}^{2/p}\frac{\Gamma(3/p)}{\Gamma(1/p)}.$

Hence, the sequence $\tilde{\mu}_{V_{n,k}X^{(n)}}$ satisfies an LDP on
$\mathcal{M}_{1}(\mathbb{R}^{k})$ with speed $n$ and rate function $I$ as
stated in Theorem 3.1. Therefore, by Lemma 4.2, $\mu_{V_{n,k}X^{(n)}}$
satisfies the same weak LDP, which extends to a full LDP by the compactness
arguments given in Lemma 4.3, thus finishing the proof. ∎

The proof of Theorem 3.2 works in a very similar way to that of Theorem 3.1,
hence we will only point out the steps where it differs from the previous
proof. Given the different scaling of $X^{(n)}$, it follows that for a Stiefel
matrix $V\in\mathbb{V}_{n,k}$, we have

$\mu_{VX^{(n)}}(A):=\mathbb{P}(VX^{(n)}\in
A)=\mathbb{P}\Big{(}\sum_{j=1}^{n}n^{\kappa/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\in A\Big{)}$ (4.7)

for any $A\in\mathcal{B}(\mathbb{R}^{k})$, and set

$\tilde{\mu}_{VX^{(n)}}(A):=\mathbb{P}\Big{(}\Big{(}\frac{1}{\beta}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}\in
A\Big{)},$ (4.8)

using the same notation as in Theorem 3.1 and its proof. The only argument
that needs to be adapted is the proof of Lemma 4.1, which will be replaced by
the following Lemma.

######  Lemma 4.4.

For $p\in[1,\infty)$ and any $n\in\mathbb{N}$ set $X^{(n)}$ as in Theorem 3.2.
Then, for $k\leq n$, we have

$\lim_{n\to\infty}\sup_{V\in\mathbb{V}_{n,k}}\rho_{\mathrm{LP}}(\mu_{VX^{(n)}},\tilde{\mu}_{VX^{(n)}})=0.$

###### Proof.

Let $A\in\mathcal{B}(\mathbb{R}^{k})$, and $\varepsilon>0$. Then, by analogue
expansion as in (4.3), we have that

$\displaystyle\tilde{\mu}_{VX^{(n)}}(A)$
$\displaystyle\leq\mu_{VX^{(n)}}(A_{\varepsilon})+\mathbb{P}\Big{(}\Big{\lVert}\Big{(}\frac{1}{\beta}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}-\sum_{j=1}^{n}n^{\kappa/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\Big{\rVert}_{2}\geq\varepsilon\Big{)}.$
(4.9)

Again, we need to show that the second summand on the right-hand side in the
above converges to $0$, as $n\to\infty$. By Markov’s inequality, it holds that

$\displaystyle\mathbb{P}\Big{(}\Big{\lVert}\Big{(}\frac{1}{\beta}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}-\sum_{j=1}^{n}n^{\kappa/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\Big{\rVert}_{2}\geq\varepsilon\Big{)}$
$\displaystyle\hskip
56.9055pt\leq\varepsilon^{-1}\mathbb{E}\Big{\lVert}\Big{(}\frac{1}{\beta}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}-\sum_{j=1}^{n}n^{\kappa/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\Big{\rVert}_{2},$

and a further application of the Cauchy–Schwarz inequality as in (4.4) yields

$\displaystyle\mathbb{E}\Big{\lVert}\Big{(}\frac{1}{\beta}\Big{)}^{1/p}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}-\sum_{j=1}^{n}n^{\kappa/p}\frac{Z_{j}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}V_{\bullet,j}\Big{\rVert}_{2}$
$\displaystyle\qquad\leq\sqrt{\mathbb{E}\Big{\lVert}\sum_{j=1}^{n}Z_{j}V_{\bullet,j}\Big{\lVert}_{2}^{2}}\cdot\sqrt{\mathbb{E}\Big{|}\Big{(}\frac{1}{\beta}\Big{)}^{1/p}-\frac{n^{\kappa/p}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}\Big{|}^{2}},$ (4.10)

with the first factor simplifying to $k\mathbb{E}[Z_{1}^{2}]$ as in (4.5). It
remains to show that

$\xi_{n}:=\Big{(}\Big{(}\frac{1}{\beta}\Big{)}^{1/p}-\frac{n^{\kappa/p}}{(\lVert
Z^{(n)}\rVert_{p}^{p}+W_{n})^{1/p}}\Big{)}^{2}\longrightarrow 0$ (4.11)

in probability, as $n\to\infty$, to address the second factor. We again do so
by showing that

$\frac{\lVert
Z^{(n)}\rVert_{p}^{p}}{n^{\kappa}}+\frac{W_{n}}{n^{\kappa}}\longrightarrow\beta$

in probability due to the continuous mapping theorem. Since $\kappa>1$, by the
same arguments as in the proof of Theorem 3.1, it follows that $\lVert
Z^{(n)}\rVert_{p}^{p}/n^{\kappa}\to 0$ and the behaviour of $W_{n}$ dominates.
By assumption, $W_{n}/n^{\kappa}\longrightarrow\beta$ in probability. In fact,
we even have $\xi_{n}\longrightarrow 0$ in $L^{1}$. Indeed, since the sequence
of random variables $(W_{n}/n^{\kappa})^{-2/p}$ is uniformly integrable by
assumption, it follows that the sequence of random variables $\xi_{n}$ is
uniformly integrable as well, which which in combination with (4.11) yields
convergence in $L^{1}$. As a consequence, the second factor in (4.10)
converges to $0$. This implies that the second summand in (4.9) converges to
$0$ uniformly in $V\in\mathbb{V}_{n,k}$. Altogether, we have proven that for
any $\varepsilon>0$,

$\tilde{\mu}_{VX^{(n)}}(A)\leq\mu_{VX^{(n)}}(A_{\varepsilon})+\varepsilon$

for $n$ sufficiently large and can prove analogously that

$\mu_{VX^{(n)}}(A)\leq\tilde{\mu}_{VX^{(n)}}(A_{\varepsilon})+\varepsilon$

for $n$ sufficiently large, thus finishing the proof. ∎

Since the rest of the proof of Theorem 3.1 does not depend on the specific
choice of $\alpha$ or the scaling of the $X^{(n)}$, the remainder of the proof
of Theorem 3.2 can proceed in the very same way. It remains only to present
the proof of Lemma 3.3.

###### Proof of Lemma 3.3.

We start by observing that

$\displaystyle\mathbb{E}\left[\Big{(}\frac{n^{\kappa}}{W_{n}}\Big{)}^{4/p}\right]=n^{4\kappa/p}{b^{a_{n}}\over\Gamma(a_{n})}\int_{0}^{\infty}x^{a_{n}-1-4/p}e^{-bx}\,\textup{d}x=n^{4\kappa/p}{b^{4/p}\over\Gamma(a_{n})}\Gamma\Big{(}a_{n}-{4\over
p}\Big{)}.$

According to the inequality [9, Equation (12)] for quotients of gamma
functions (applied with $x=a_{n}+5-4/p$ and $y=4/p$) one has that

$\displaystyle{\Gamma(a_{n}-{4\over p})\over\Gamma(a_{n})}$
$\displaystyle=\Big{(}\prod_{i=0}^{4}{(a_{n}+i)\over(a_{n}-{4\over
p}+i)}\Big{)}{\Gamma(a_{n}-{4\over p}+5)\over\Gamma(a_{n}+5)}$
$\displaystyle=\Big{(}{1\over\prod_{i=0}^{4}(1-{4\over
p(a_{n}+i)})}\Big{)}{1\over{\Gamma(a_{n}-{4\over p}+5)\over\Gamma(a_{n}+5)}}$
$\displaystyle\leq\Big{(}{1\over\prod_{i=0}^{4}(1-{4\over
p(m+i)})}\Big{)}{1\over(a_{n}+4-{4\over p})^{4/p}}$
$\displaystyle\leq\Big{(}{1\over\prod_{i=0}^{4}(1-{4\over
p(m+i)})}\Big{)}{1\over a_{n}^{4/p}}=M_{p}\cdot{1\over a_{n}^{4/p}},$

where we also used that $p\geq 1$. By our assumption on the growth of $a_{n}$
and since $a_{n}$ is increasing it follows that

$\mathbb{E}\left[\Big{(}\frac{n^{\kappa}}{W_{n}}\Big{)}^{4/p}\right]\leq
n^{4\kappa/p}b^{4/p}M_{p}a_{n}^{-4/p}\leq n^{4\kappa/p}b^{4/p}M_{p}(\lambda
mn^{\kappa})^{-4/p}=b^{4/p}M_{p}(\lambda m)^{-4/p}$

for all $n\in\mathbb{N}$. This completes the proof. ∎

## References

  * [1] D. Alonso-Gutiérrez, J. Prochno, and C. Thäle, _Large deviations for high-dimensional random projections of $\ell_{p}^{n}$-balls_, Adv. in Appl. Math. 99 (2018), 1–35.
  * [2] F. Barthe, O. Guédon, S. Mendelson, and A. Naor, _A probabilistic approach to the geometry of the $\ell^{n}_{p}$-ball_, Ann. Probab. 33 (2005), no. 2, 480–513.
  * [3] D. Chafaï, O. Guédon, G. Lecué, and A. Pajor, _Interactions between Compressed Sensing Random Matrices and High Dimensional Geometry_ , Société Mathématique de France, 2012.
  * [4] A. Dembo and O. Zeitouni, _Large Deviations. Techniques and Applications_ , Stochastic Modelling and Applied Probability, vol. 38, Springer-Verlag, Berlin, 2010, Corrected reprint of the second (1998) edition.
  * [5] S. Foucart and H. Rauhut, _A Mathematical Introduction to Compressive Sensing_ , Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2013.
  * [6] N. Gantert, S. S. Kim, and K. Ramanan, _Large deviations for random projections of $\ell^{p}$ balls_, Ann. Probab. 45 (2017), no. 6B, 4419–4476.
  * [7] A. Hinrichs, D. Krieg, E. Novak, J. Prochno, and M. Ullrich, _Random sections of ellipsoids and the power of random information_ , Trans. Am. Math. Soc. 374 (2021), no. 12, 8691–8713.
  * [8] A. Hinrichs, J. Prochno, and M. Sonnleitner, _Random sections of $\ell_{p}$-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators_, arXiv:2109.14504.
  * [9] G.J.O. Jameson, _Inequalities for gamma function ratios_ , Amer. Math. Monthly 120 (2013), no. 10, 936–940.
  * [10] Z. Kabluchko and J. Prochno, _Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions_ , arXiv:2110.12977.
  * [11] Z. Kabluchko, J. Prochno, and C. Thäle, _High-dimensional limit theorems for random vectors in $\ell_{p}^{n}$-balls_, Commun. Contemp. Math. 21 (2019), no. 1, 1750092, 30pp.
  * [12] Z. Kabluchko, J. Prochno, and C. Thäle, _Sanov-type large deviations in Schatten classes_ , Ann. Inst. H. Poincaré Probab. Statist. 56 (2019), no. 2, 928–953.
  * [13] Zakhar Kabluchko, Joscha Prochno, and Christoph Thäle, _High-dimensional limit theorems for random vectors in $\ell^{n}_{p}$-balls. II_, Commun. Contemp. Math. 23 (2021), no. 3, 1950073, 35pp.
  * [14] T. Kaufmann and C. Thäle, _Weighted $p$-radial distributions on Euclidean and matrix $p$-balls with applications to large deviations_, arXiv:2109.01370.
  * [15] S. S. Kim and K. Ramanan, _Large deviation principles induced by the Stiefel manifold, and random multi-dimensional projections_ , arXiv:2105.04685.
  * [16] B. Klartag, _A central limit theorem for convex sets_ , Invent. Math. 168 (2007), no. 1, 91–131.
  * [17] , _Power-law estimates for the central limit theorem for convex sets_ , J. Funct. Anal. 245 (2007), no. 1, 284–310.
  * [18] D. Krieg and M. Sonnleitner, _Random points are optimal for the approximation of Sobolev functions_ , arXiv:2009.11275.
  * [19] J. Prochno, C. Thäle, and N. Turchi, _Geometry of $\ell_{p}^{n}$-balls: Classical results and recent developments_, High Dimensional Probability VIII, Progress in Probability. N. Gozlan, R. Latala, K. Loucini, M. Madiman eds., Birkhäuser, 2019.