Large deviation principles induced by the Stiefel manifold, and random multi-
dimensional projections
Steven Soojin Kim111These results are part of this author’s Ph.D. thesis [15]
and Kavita Ramanan222Kavita Ramanan was supported in part by NSF-DMS Grants
1713032 and 1954351
Brown University

This article is dedicated to the memory of Elizabeth Meckes

###### Abstract

For fixed positive integers $k<n$, given an $n$-dimensional random vector
$X^{(n)}$, consider its $k$-dimensional projection $\mathbf{a}_{n,k}X^{(n)}$,
where $\mathbf{a}_{n,k}$ is an $n\times k$-dimensional matrix belonging to the
Stiefel manifold $\mathbb{V}_{n,k}$ of orthonormal $k$-frames in
$\mathbb{R}^{n}$. For a class of sequences $\\{X^{(n)}\\}_{n\in\mathbb{N}}$
that includes the uniform distributions on suitably scaled $\ell_{p}^{n}$
balls, $p\in(1,\infty]$, and product measures with sufficiently light tails,
it is shown that the sequence of projected vectors
$\\{\mathbf{a}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}}$ satisfies a large
deviation principle whenever the empirical measures of the rows of
$\sqrt{n}\mathbf{a}_{n,k}$ converge, as $n\rightarrow\infty$, to a probability
measure on $\mathbb{R}^{k}$. In particular, when $\mathbf{A}_{n,k}$ is a
random matrix drawn from the Haar measure on $\mathbb{V}_{n,k}$, this is shown
to imply a large deviation principle for the sequence of random projections
$\\{\mathbf{A}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}}$ in the quenched
sense (that is, conditioned on almost sure realizations of
$\\{\mathbf{A}_{n,k}\\}_{n\in\mathbb{N}}$). Moreover, a variational formula is
obtained for the rate function of the large deviation principle for the
annealed projections
$\\{\mathbf{A}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}}$, which is
expressed in terms of a family of quenched rate functions and a modified
entropy term. A key step in this analysis is a large deviation principle for
the sequence of empirical measures of rows of $\sqrt{n}\mathbf{A}_{n,k}$,
$n\geq k$, which may be of independent interest. The study of multi-
dimensional random projections of high-dimensional measures is of interest in
asymptotic functional analysis, convex geometry and statistics. Prior results
on quenched large deviations for random projections of $\ell_{p}^{n}$ balls
have been essentially restricted to the one-dimensional setting.

Keywords: large deviations; Stiefel manifold; random projections; quenched;
annealed; rate function; $\ell_{p}^{n}$ balls; asymptotic convex geometry;
variational formula

MSC 2020 subject classifications: 60F10; 60B20; 52A23

##  1 Introduction

###  1.1 Background

The study of high-dimensional measures and their lower-dimensional projections
is a central theme in high-dimensional probability, asymptotic functional
analysis and convex geometry, where in the latter case the measures of
interest are distributions on convex bodies, which are compact, convex sets
with non-empty interior (see, e.g., [16, 22]). Multidimensional projections of
high-dimensional random vectors are also relevant in statistics, data analysis
and computer science [7, 13]. Recent work has shown that large deviation
principles (LDPs) that capture the tail behavior of lower-dimensional random
projections can provide more interesting information about the original high-
dimensonal measures than central limit theorem type results that capture
universal phenomena of fluctuations. For example, in the case of
$\ell_{p}^{n}$ balls, $p\in[1,\infty)$, which are fundamental objects in
convex geometry, this was first illustrated by LDPs for one-dimensional
projections obtained in [10, 15], and subsequently by LDPs for norms of
samples from $\ell_{p}^{n}$ balls and their multi-dimensional projections in
[1, 18, 17], as well as corresponding refined large deviation estimates
obtained in [21, 14]. LDPs of random projections of high-dimensional measures
are broadly of two types, the terminology arising from statistical physics:
so-called “quenched” LDPs, where one conditions on the choice of the (sequence
of) sub-spaces, bases or directions onto which one projects; or “annealed”
LDPs, which average over the randomness arising in the choice of the
projection. While most of the work described above on $\ell_{p}^{n}$ balls
focused on one-dimensional LDPs (either for one-dimensional projections or
norms of higher-dimensional projections), in [17], annealed LDPs were also
established for multi-dimensional projections of high-dimensional measures
that satisfy a general condition called the asymptotic thin shell condition.
This condition was shown to be satisfied in [17] by several classes of
measures, including product measures with polynomial tail decay,
$\ell_{p}^{n}$ balls, $p\in[1,\infty]$, and classes of Orlicz balls and Gibbs
measures.

In this article, we establish quenched LDPs for multidimensional random
projections of a class of high-dimensional measures as the dimension $n$ goes
to infinity. Quenched LDPs can often provide more geometric information than
annealed LDPs, but their analysis is typically more difficult because one can
no longer exploit symmetry properties of the random projection measure. To
state our results more precisely, for $k\in\mathbb{R}^{n}$, let $I_{k}$ denote
the $k\times k$ identity matrix, and for $n>k$, let

$\mathbb{V}_{n,k}:=\\{A\in\mathbb{R}^{n\times k}:A^{\intercal}A=I_{k}\\}$
(1.1)

denote the Stiefel manifold of $k$-frames in $\mathbb{R}^{n}$. Observe that
the set $\mathbb{V}_{n,n}$ can be identified with the set ${\mathcal{O}}(n)$
of $n\times n$ orthogonal matrices with columns of norm $1$. Also, note that
for $k,n\in\mathbb{N}$, $k<n$, any $a_{n,k}\in\mathbb{V}_{n,k}$ defines a
linear projection from $n$ to $k$ dimensions. Fixing a probability space
$(\Omega,\mathcal{F},\mathbb{P})$, we consider sequences of random vectors
$\\{X^{(n)}\\}_{n\in\mathbb{N}}$ defined on this space that satisfy a certain
set of conditions (see Assumption 2.1), which include, for example, $X^{(n)}$
uniformly distributed on an $\ell_{p}^{n}$ ball of radius $n^{1/p}$, $p\geq
2$, or $X^{(n)}$ distributed according to a product measure with sufficiently
light tails. For any fixed $k\in\mathbb{N}$, let
$\mathbb{N}_{k}:=\\{n\in\mathbb{N}:n>k\\}$, and consider the sequence of
$k$-dimensional projections

$\\{n^{-1/2}\mathbf{a}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}_{k}},$
(1.2)

where $\mathbf{a}:=\\{\mathbf{a}_{n,k}\\}_{n\in\mathbb{N}_{k}}$, with
$\mathbf{a}_{n,k}\subset\mathbb{V}_{n,k}$ for each $n\in\mathbb{N}_{k}$. Also,
let

$\mathsf{L}_{n,k}^{\mathbf{a}}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{\sqrt{n}\mathbf{a}_{n,k}(i,\cdot\,)},\quad
n\in\mathbb{N}_{k},$ (1.3)

be the associated sequence of empirical measures of the rows of
$\sqrt{n}\mathbf{a}_{n,k}$. Our first result, Theorem 2.4, shows that whenever
$\\{\mathsf{L}_{n,k}^{\mathbf{a}}\\}_{n\in\mathbb{N}_{k}}$ converges in the
$q_{\star}$-Wasserstein topology (see Definition 1.2) to a measure $\nu$, then
the sequence of random projections
$\\{n^{-1/2}\mathbf{a}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}_{k}},$
satisfies an LDP with a rate function that we denote by
${\mathcal{J}}^{\sf{qu}}_{\nu}$. In particular, this implies a quenched LDP
for the sequence
$\\{\mathbf{A}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}_{k}},$ where the
random matrix

$\mathbf{A}_{n,k}=[\mathbf{A}_{n,k}(i,j)]_{i=1,\dots,n;\,j=1,\dots,k}$

is sampled, independently of $\\{X^{(n)}\\}_{n\in\mathbb{N}}$, from
$\sigma_{n,k}$, the Haar measure on $\mathbb{V}_{n,k}$ (i.e., the unique
probability measure on $\mathbb{V}_{n,k}$ that is invariant under the group
${\mathcal{O}}(n)$ of orthogonal transformations). In [17], it was shown that
$\\{\mathbf{A}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}_{k}}$ also
satisfies an annealed LDP. Our second result, Theorem 2.7, establishes a
variational formula for the (annealed) rate function
${\mathcal{J}}^{\sf{an}}$, in terms of the quenched rate functions
${\mathcal{J}}^{\sf{qu}}_{\nu}$. Along the way, for any $q\in(0,2)$, in
Theorem 2.8, we also establish an LDP for the random empirical measure
sequence $\\{\mathsf{L}^{A}_{n,k}\\}_{n\in\mathbb{N}_{k}}$ in the
$q$-Wasserestein topology, which may be of independent interest.

In the next section, we introduce some basic notation and terminology that
will be used throughout, and then provide precise statements of our main
results in Section 2, with proofs presented in Sections 3–6.

###  1.2 Notation and Terminology

We first recall the definition of an LDP, referring to [8] for further
background on large deviations theory.

######  Definition 1.1.

Let $\mathcal{X}$ be a topological space with Borel sigma-algebra
$\mathcal{B}$. A sequence of probability measures
$\\{P_{n}\\}_{n\in\mathbb{N}}\subset\mathcal{P}(\mathcal{X})$ is said to
satisfy a _large deviation principle (LDP)_ in $\mathcal{X}$ with rate
function $I:\mathcal{X}\rightarrow[0,\infty]$ if for all $B\in\mathcal{B}$,

$-\inf_{x\in B^{\circ}}I(x)\leq\liminf_{n\rightarrow\infty}\tfrac{1}{n}\log
P_{n}(B)\leq\limsup_{n\rightarrow\infty}\tfrac{1}{n}\log
P_{n}(B)\leq-\inf_{x\in\bar{B}}I(x),$

where $B^{\circ}$ and $\bar{B}$ denote the interior and closure of $B$,
respectively. We say $I$ is a _good rate function (GRF)_ if it has compact
level sets. Analogously, a sequence of $\mathcal{X}$-valued random variables
$\\{\mathsf{x}_{n}\\}_{n\in\mathbb{N}}$ is said to satisfy an LDP with GRF $I$
if the sequence of their laws
$\\{\mathbb{P}\circ\mathsf{x}_{n}\\}_{n\in\mathbb{N}}$ does.

We recall some definitions that will prove useful in our discussion of rate
functions. Given a function $f:\mathbb{R}^{m}\rightarrow[-\infty,\infty]$ for
some $m\in\mathbb{N}$, we let $f^{*}$ denote its _Legendre transform_ :

$f^{*}(t):=\sup_{s\in\mathbb{R}^{m}}\left[\langle
s,t\rangle-f(s)\right],\qquad t\in\mathbb{R}^{m}.$

Since we will frequently invoke the contraction principle, Cramér’s theorem
and Sanov’s theorem, we refer the reader to Theorem 4.2.1, Corollary 6.1.6,
and Section 6.2, respectively, of [8], for precise statements.

Next, for $p\in[1,\infty]$, let $\|\cdot\|_{p}$ denote the $\ell_{p}^{k}$ norm
on $\mathbb{R}^{k}$. When $p=2$, and where the context makes it clear, we will
omit the subscript and simply write $\|\cdot\|$ for the Euclidean norm. Let
$\mathcal{P}(\mathbb{R}^{k})$ denote the space of probability measures on
$\mathbb{R}^{k}$, by default equipped with the topology of weak convergence.
We will sometimes consider the following restricted subsets of probability
measures: for $q>0$, let

$\mathcal{P}_{q}(\mathbb{R}^{k}):=\left\\{\nu\in\mathcal{P}(\mathbb{R}^{k}):\int_{\mathbb{R}^{k}}\|x\|^{q}\nu(dx)<\infty\right\\}.$

######  Definition 1.2.

For $q>0$, we say a sequence of probability measures
$\\{\nu_{n}\\}_{n\in\mathbb{N}}\subset\mathcal{P}_{q}(\mathbb{R}^{d})$
converges to a limit $\nu$ with respect to the _$q$ -Wasserstein topology_ if
we have both weak convergence, denoted $\nu_{n}\Rightarrow\nu$, as well as
convergence of $q$-th moments
$\int_{\mathbb{R}^{d}}\|x\|^{q}\nu_{n}(dx)\rightarrow\int_{\mathbb{R}^{d}}\|x\|^{q}\nu(dx)$.
As noted in Section 6 of [30], the $q$-Wasserstein topology is metrizable
through the $q$-Wasserstein metric which we denote by $\mathcal{W}_{q}$.

##  2 Main results

We now provide a precise statement of our results. We start by defining the
class of random vectors that we consider. As in [8, Definition 2.3.5], we
define the domain of an (extended real-valued) function $f$ defined on a
Euclidean space $S$, denoted $D_{f}$, as the subset of points in $S$ for which
$f$ is finite; furthermore, the function $f$ is said to be _essentially
smooth_ if: $D_{f}\neq\emptyset$; $f$ is differentiable in the interior
$D_{f}^{\circ}$ of $D_{f}$; and $f$ is “steep” (i.e., if $D_{f}$ has a
boundary $\partial D_{f}$, then $\lim_{t\rightarrow\partial D_{f}}|\nabla
f(t)|=\infty$).

######  Assumption 2.1 (quenched).

We impose the following conditions on the sequence of random vectors
$\\{X^{(n)}\\}_{n\in\mathbb{N}}$.

  1. (i)

Representation: there exists a sequence of i.i.d. real-valued random variables
$\\{\xi_{j}\\}_{j\in\mathbb{N}}$, a Borel measurable function
$\mathbf{r}:\mathbb{R}\rightarrow\mathbb{R}_{+}$, and a continuous function
$\rho:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ such that

$X^{(n)}\buildrel(d)\over{=}\xi^{(n)}\cdot\rho\left(\frac{1}{n}\sum_{i=1}^{n}\mathbf{r}(\xi_{i})\right),\quad
n\in\mathbb{N},$

where $\xi^{(n)}:=(\xi_{1},\dots,\xi_{n})$. Let $\Lambda$ denote the log
moment-generating function (mgf) of $(\xi_{1},\mathbf{r}(\xi_{1}))$:

$\Lambda(s_{1},s_{2}):=\log\mathbb{E}\left[\exp\left(s_{1}\xi_{1}+s_{2}\mathbf{r}(\xi_{1})\right)\right],\quad
s_{1}\in\mathbb{R},s_{2}\in\mathbb{R}.$ (2.1)

  2. (ii)

Log Moment-Generating Function (MGF): There exists $q_{\star}>0$ such that for
every $(s_{1},s_{2})\in D_{\Lambda},$ there exists a finite constant
$C_{s_{2}}$ (depending only on $s_{2}$ and not $s_{1}$) such that

$\Lambda(s_{1},s_{2})\leq C_{s_{2}}(1+|s_{1}|^{q_{\star}}).$ (2.2)

Furthermore, there exists $T\leq\infty$ such that
$D_{\Lambda}=\mathbb{R}\times(-\infty,T)$.

  3. (iii)

Integrated Log Mgf: For any $\nu\in\mathcal{P}(\mathbb{R}^{k})$, the function
$\Psi_{\nu}:\mathbb{R}^{k+1}\rightarrow\mathbb{R}$ obtained as an integrated
form of the log mgf,

$\Psi_{\nu}(t_{1},t_{2}):=\int_{\mathbb{R}^{k}}\Lambda\left(\langle
t_{1},x\rangle,t_{2}\right)\nu(dx),\quad
t_{1}\in\mathbb{R}^{k},t_{2}\in\mathbb{R},$ (2.3)

contains 0 in the interior of its domain, is lower semicontinuous, and is
essentially smooth.

  4. (iv)

Log MGF of Square: The log mgf $\bar{\Lambda}$ of
$(\xi_{1}^{2},\mathbf{r}(\xi_{1}))$, given by

$\bar{\Lambda}(s_{1},s_{2}):=\log\mathbb{E}[\exp(s_{1}\xi_{1}^{2}+s_{2}\mathbf{r}(\xi_{1}))],\qquad
s_{1}\in\mathbb{R},\,s_{2}\in\mathbb{R},$ (2.4)

is finite in a non-empty neighborhood of the origin $(0,0)$.

  5. (v)

Tail Bound: The exponent $q_{\star}$ of part (ii) is bounded above,
$q_{\star}<2$.

######  Remark 2.2.

The inequality (2.2) in Assumption 2.1(ii) implies that for
$t_{1}\in\mathbb{R}^{k}$ and $t_{2}\in(-\infty,T)$, the map
$\nu\mapsto\Psi_{\nu}(t_{1},t_{2})$ is continuous with respect to the
$q_{\star}$-Wasserstein topology. Further, for $t_{2}>T$, we have
$\Lambda(y,t_{2})=\infty$ for all $y\in\mathbb{R}$, and hence, (2.3) shows
that $\Psi(t_{1},t_{2})=\infty$ for all $t_{1}\in\mathbb{R}$.

######  Remark 2.3.

A wide class of product measures satisfy Assumption 2.1 with
$\rho\equiv\mathbf{r}\equiv 1$; namely those that have sufficiently light
tails, in the sense of parts (iv) and (v). Examples of sequences of non-
product measures satisfying Assumption 2.1 are $\ell_{p}^{n}$ spheres. More
precisely, fix $p\in[1,\infty)$, and for $n\in\mathbb{N}$, let
$\mathbb{D}_{n,p}:=\\{x\in\mathbb{R}^{n}:\sum_{i=1}^{n}|x_{i}|^{p}=n\\}$ be
the scaled $\ell_{p}^{n}$ ball in $\mathbb{R}^{n}$, let
$\mathbb{S}_{n,p}:=\partial\mathbb{D}_{n,p}$ be the scaled $\ell_{p}^{n}$
sphere in $\mathbb{R}^{n}$, let $\eta_{n,p}$ be the cone measure on
$\mathbb{S}_{n,p}$: for Borel subsets $S\subset\mathbb{S}_{n,p}$,

$\eta_{n,p}(S):=\frac{{\rm vol}_{n}(\\{cx:x\in S,c\in[0,n^{1/p}]\\})}{{\rm
vol}_{n}(\mathbb{D}_{n,p})},$

with ${\rm vol}_{n}$ denoting Lebesgue measure on $\mathbb{R}^{n}$, and let
$X^{(n)}=X^{(n,p)}$ be distributed according to $\eta_{n,p}$. Then:

  1. (i)

for $p\in[1,\infty)$, this condition follows from results in [28, 26], with
$\\{\xi_{j}\\}_{j\in\mathbb{N}}$ being the i.i.d. sequence with common law
equal to the generalized $p$-normal distribution (namely, the probability
measure on $\mathbb{R}$ with density proportional to $e^{-|y|^{p}/p}$),
$\mathbf{r}(x)=|x|^{p}$, and $\rho(y)=y^{-1/p}$;

  2. (ii)

for $p\in(1,\infty)$, the growth conditions on the log mgf $\Lambda$ are
established in [10, Lemma 5.7]; further, $\Lambda$ is symmetric in its first
argument due to the symmetry of the generalized $p$-normal distribution;

  3. (iii)

for $p\in(1,\infty)$, the conditions on the integrated log mgf are established
in [10, Lemma 5.9];

  4. (iv)

for $p\in[2,\infty)$, the log mgf condition is easily verified;

  5. (v)

for $p\in(2,\infty)$, the precise tail bound exponent is established in [10,
Lemma 5.5].

We now our state our first result, whose proof is deferred to Section 5.
Recall the $q$-Wasserstein metric $\mathcal{W}_{q}$ specified in Definition
1.2. Also, for any $\nu\in\mathcal{P}(\mathbb{R}^{k})$, we let
$\Psi_{\nu}^{*}$ denote the Legendre transform of $\Psi_{\nu}$,

$\Psi_{\nu}^{*}(\tau_{1},\tau_{2}):=\sup_{t_{1}\in\mathbb{R}^{k},t_{2}\in\mathbb{R}}\\{\langle
t_{1},\tau_{1}\rangle+t_{2}\tau_{2}-\Psi_{\nu}(t_{1},t_{2})\\},\quad\tau_{1}\in\mathbb{R}^{k},\tau_{2}\in\mathbb{R}.$
(2.5)

Also, let $\gamma$ denote the standard Gaussian distribution on $\mathbb{R}$,
and $\gamma^{\otimes k}$ its $k$-fold product.

######  Theorem 2.4 (Quenched).

Fix $k\in\mathbb{N}$, and suppose $\\{X^{(n)}\\}_{n\in\mathbb{N}}$ satisfies
Assumption 2.1(i ii, iii) with associated $q_{\star}>0$ and $\Psi_{\nu}$.
Choose any sequence
$\mathbf{a}=\\{\mathbf{a}_{n,k}\\}_{n\in\mathbb{N}_{k}}\subset\mathbb{V}_{n,k}$
such that, for some $\nu\in\mathcal{P}(\mathbb{R}^{k})$,

$\mathcal{W}_{q_{\star}}(\mathsf{L}_{n,k}^{\mathbf{a}},\nu)\to 0$ (2.6)

where $\mathsf{L}_{n,k}^{\mathbf{a}}\in\mathcal{P}(\mathbb{R}^{k})$ is the
empirical measure defined in (1.3). Then, the following claims hold:

  1. (i)

The sequence
$\\{n^{-1/2}\mathbf{a}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}_{k}}$
satisfies an LDP in $\mathbb{R}^{k}$ with GRF
$\mathcal{J}_{\nu}^{\sf{qu}}:\mathbb{R}^{k}\rightarrow[0,\infty]$ defined by

$\mathcal{J}_{\nu}^{\sf{qu}}(x):=\inf_{\tau\in\mathbb{R}_{+}}\Psi_{\nu}^{*}\left(\tfrac{x}{\rho(\tau)},\tau\right),\quad
x\in\mathbb{R}^{k}.$ (2.7)

  2. (ii)

If $\sigma$ is any probability measure on
$\mathbb{S}:=\otimes_{n>k}\mathbb{V}_{n,k}$ whose $n$-th marginal coincides
with the Haar measure $\sigma_{n,k}$, then for $\sigma$-a.e.
$\mathbf{a}=\\{\mathbf{a}_{n,k}\\}_{n\in\mathbb{N}_{k}}\in\mathbb{S}$, the
sequence
$\\{n^{-1/2}\mathbf{a}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}_{k}}$
satisfies an LDP in $\mathbb{R}^{k}$ with GRF $\mathcal{J}_{\gamma^{\otimes
k}}^{\sf{qu}}$.

  3. (iii)

Let $U$ be a uniformly distributed random variable on $[0,1]$, independent of
$\\{X^{(n)}\\}_{n\in\mathbb{N}}$. If the log mgf $\Lambda$ of (2.1) is
symmetric in its first argument, then the claims (i) and (ii) also hold for
the sequence
$\\{n^{-1/2}\mathbf{a}_{n,k}^{\intercal}X^{(n)}U^{1/n}\\}_{n\in\mathbb{N}_{k}}$.

######  Remark 2.5.

Claim (iii) of Theorem 2.4 is motivated by the observation that if $X^{(n,p)}$
is distributed according to the cone measure on the scaled $\ell_{p}^{n}$
sphere $\mathbb{S}_{n,p}$, then the random variable $U^{1/n}X^{(n,p)}$ is
uniformly distributed on the scaled $\ell_{p}^{n}$ ball $\mathbb{D}_{n,p}$
[28]. Since, as noted in Remark 2.3, $X^{(n,p)}$ satisfies Assumption 2.1
across a wide range of $p$ (with symmetric $\Lambda$), Theorem 2.4(iii) allows
an extension of the LDP results in (i) and (ii) of Theorem 2.4 from
$\ell_{p}^{n}$ spheres to $\ell_{p}^{n}$ balls, which are of greater interest
in convex geometry.

Note that the rate function $\mathcal{J}_{\nu}^{\sf{qu}}$ depends only on the
limit $\nu$ in (2.6), and is insensitive to further specifics of the
projection matrix sequence $\mathbf{a}$. For one-dimensional projections
($k=1$), Theorem 2.4 recovers both Theorem 2 of [9], which addresses the case
where $X^{(n)}$ has a product distribution, and Theorem 2.5 and Proposition
5.3 of [10], which consider the case when $X^{(n)}$ is uniformly distributed
on $\mathbb{D}_{n,p}$ or according to the cone measure $\eta_{n,p}$ (as
defined in Remark 2.2). One setting of multidimensional projections ($k>1$)
considered prior to the above result is the LDP for the projection of
$X^{(n)}$ onto the first $k$ canonical directions, where $\mathbf{a}_{n,k}$ is
the matrix of 1s on the diagonal and 0s elsewhere, which does not satisfy
(2.6). More recent work [19] establishes interesting asymptotics (law of large
numbers and LDPs) for the shape of multidimensional projections of the uniform
distribution on a cube or discrete cube. This paper differs by establishing
almost everywhere quenched LDP results, first reported in the PhD thesis [15],
for multidimensional projections beyond the particular cases of the canonical
projection and product measures. Our results provide a potential starting
point for obtaining asymptotic results for shapes and instrinsic volumes of
projections of non-product measures such as $\ell_{p}^{n}$ balls, as well as
for ongoing work on sharp quenched large deviation estimates for multi-
dimensional projections and their norms, which are relevant for understanding
volumetric properties of convex bodies and their intersections.

Our second main result concerns a variational representation of the annealed
rate function for the sequence of random multi-dimensional projections
$\\{\mathbf{A}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}_{k}}$. We start by
stating an annealed LDP counterpart to Theorem 2.4, specialized to the setting
considered in this article.

######  Theorem 2.6 (Annealed).

Consider a sequence $\\{X^{(n)}\\}_{n\in\mathbb{N}}$ that satisfies Assumption
2.1(i, iv) with associated $\\{\xi_{j}\\}_{j\in\mathbb{N}}$, $\rho$, and
$\bar{\Lambda}$. Then, for any $k\in\mathbb{N}$,
$\\{n^{-1/2}\mathbf{A}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}_{k}}$
satisfies an LDP in $\mathbb{R}^{k}$ with GRF
$\mathcal{J}^{\sf{an}}:\mathbb{R}^{k}\rightarrow[0,\infty]$ defined by

$\mathcal{J}^{\sf{an}}(x):=\inf_{c>0}\left\\{J_{X}\left(\tfrac{\|x\|_{2}}{c}\right)-\tfrac{1}{2}\log(1-c^{2})\right\\},\quad
x\in\mathbb{R}^{k},$

where $J_{X}$ is given, in terms of the Legendre transform $\bar{\Lambda}^{*}$
of $\bar{\Lambda}$, by

$J_{X}(x):=\inf_{(t_{1},t_{2})\in\mathbb{R}_{+}^{2}}\left\\{\bar{\Lambda}^{*}(t_{1},t_{2}):x=t_{1}^{1/2}\rho(t_{2})\right\\}=\inf_{t_{2}>0}\bar{\Lambda}^{*}\left(\frac{x^{2}}{\rho^{2}(t_{2})},t_{2}\right).$

###### Proof.

It follows from Theorem 2.7 of [17] that
$\\{n^{-1/2}\mathbf{A}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}_{k}}$
satisfies an LDP in $\mathbb{R}^{k}$ with GRF $\mathcal{J}^{\sf{an}}$ as
defined above whenever Assumption A* therein is satisfied with speed
$s_{n}=n$, namely, when the sequence of scaled norms
$\\{\|X^{(n)}\|_{2}/\sqrt{n}\\}_{n\in\mathbb{N}}$ satisfies an LDP with GRF
$J_{X}$. Since the domain of $\bar{\Lambda}$ contains a neighborhood of the
origin due to Assumption 2.1(iv), Cramér’s theorem implies that the sequence
$\\{\left(\frac{1}{n}\sum_{i=1}^{n}\xi_{i}^{2},\frac{1}{n}\sum_{i=1}^{n}\mathbf{r}(\xi_{i})\right)\\}_{n\in\mathbb{N}}$
satisfies an LDP in $\mathbb{R}^{2}$ with GRF $\bar{\Lambda}^{*}$. Since
Assumption 2.1(i) implies

$\frac{\|X^{(n)}\|_{2}}{\sqrt{n}}\buildrel(d)\over{=}\left(\frac{1}{n}\sum_{i=1}^{n}\xi_{i}^{2}\right)^{1/2}\rho\left(\frac{1}{n}\sum_{i=1}^{n}\mathbf{r}(\xi_{i})\right),$

with $\rho$ continuous, the contraction principle shows that the sequence of
scaled norms $\\{\|X^{(n)}\|_{2}/\sqrt{n}\\}_{n\in\mathbb{N}}$ satisfies an
LDP with GRF $J_{X}$. This completes the proof. ∎

To state the variational representation for the rate function
$\mathcal{J}^{\sf{an}}$, we first introduce some notation. For
$\nu,\mu\in\mathcal{P}(\mathbb{R})$, define the _relative entropy_ of $\nu$
with respect to $\mu$ as

$H(\nu|\mu):=\int_{\mathbb{R}}\log\left(\tfrac{d\nu}{d\mu}\right)d\nu$ (2.8)

if $\nu\ll\mu$, and $H(\nu|\mu):=+\infty$ otherwise. Let $\gamma$ denote the
standard Gaussian measure on $\mathbb{R}$, and $\gamma^{\otimes k}$ the
associated product measure on $\mathbb{R}^{k}$. For
$\nu\in\mathcal{P}(\mathbb{R}^{k})$, let
$\mathcal{C}:\mathcal{P}(\mathbb{R}^{k})\rightarrow\mathbb{R}^{k\times k}$
denote the covariance map,

$\mathcal{C}(\nu):=\int_{\mathbb{R}^{k}}[x\otimes
x]\,\nu(dx),\quad\nu\in\mathcal{P}(\mathbb{R}^{k}).$ (2.9)

Let $I_{k}$ denote the $k\times k$ identity matrix, write $A\preceq B$ if
$B-A$ is positive semidefinite, and define the rate function

$\mathbb{H}_{k}(\nu):=\left\\{\begin{array}[]{ll}H(\nu|\gamma^{\otimes
k})+\frac{1}{2}\textnormal{tr}(I_{k}-\mathcal{C}(\nu))&\textnormal{ if
}\mathcal{C}(\nu)\preceq I_{k}\\\ +\infty&\textnormal{
else}\end{array}\right.,\quad\nu\in\mathcal{P}(\mathbb{R}^{k}).$ (2.10)

######  Theorem 2.7.

Fix $k\in\mathbb{N}$, suppose that the sequence
$\\{X^{(n)}\\}_{n\in\mathbb{N}}$ satisfies Assumption 2.1. Let
$\mathcal{J}_{\nu}^{\sf{qu}}$ and $\mathcal{J}^{\sf{an}}$ be defined as in
Theorems 2.4 and 2.6, respectively. Then, we have the following variational
formula:

$\mathcal{J}^{\sf{an}}(x)=\inf_{\nu\in\mathcal{P}(\mathbb{R}^{k})}\left\\{\mathcal{J}_{\nu}^{\sf{qu}}(x)+\mathbb{H}_{k}(\nu)\right\\},\qquad
x\in\mathbb{R}^{k}.$ (2.11)

Note that $\mathbb{H}_{k}(\nu)=0$ when $\nu=\gamma^{\otimes k}$, which implies
$\mathcal{J}^{\sf{an}}\leq\mathcal{J}^{\sf{qu}}_{\gamma^{\otimes k}}$, as
would be expected from Jensen’s inequality given
$\mathcal{J}^{\sf{qu}}_{\gamma^{\otimes k}}$ is simply the GRF of the quenched
LDP for $\\{\mathbf{A}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}_{k}}$. More
generally, the optimization problem (2.11) can be interpreted as saying that
at the large deviation level, the annealed probability of a rare event is the
infimum, over all random “environments” (in this case “projections”), of the
probability of the rare event conditioned on that environment plus the cost of
the choice of the environment. While such a relation is intuitive, rigorous
proofs of such informal statements are typically non-trivial. For example,
such variational representations have been rigorously established only in a
few specific cases, such as LDPs for random walks in random environments on
$\mathbb{Z}$ in [6] and on supercritical Galton-Watson trees in [2]. The one-
dimensional case ($k=1$) of Theorem 2.7 for $\ell_{p}^{n}$ balls recovers
Theorem 2.7 of [10]. The proof of the the multi-dimensional case stated in
Theorem 2.7), which is given in Section 6, is more involved and relies on an
auxiliary LDP for the following sequence of random empirical measures,
analogous to those defined in (1.3):

$\mathsf{L}_{n,k}:=\mathsf{L}_{n,k}^{\mathbf{A}}=\frac{1}{n}\sum_{i=1}^{n}\delta_{\sqrt{n}\mathbf{A}_{n,k}(i,\cdot)},\quad
n\in\mathbb{N}.$ (2.12)

######  Theorem 2.8.

Fix $k\in\mathbb{N}$. Then for all $q\in(0,2)$, the sequence
$\\{\mathsf{L}_{n,k}\\}_{n\in\mathbb{N}_{k}}$ satisfies an LDP in
$\mathcal{P}_{q}(\mathbb{R}^{k})$ with the strictly convex GRF
$\mathbb{H}_{k}$.

This theorem, which is established in Section 4, generalizes Theorem 6.6 of
[3], which states an LDP for the empirical measure of coordinates drawn
uniformly from the sphere $S^{n-1}$, which corresponds to the case $k=1$ in
our work. In contrast to this case, the $k>1$ case necessitates more extensive
computations which arise due to the non-commutative matrix setting, where the
Bartlett decomposition of Proposition 3.2 replaces the usual polar
decomposition for a random vector from the sphere. Given that large deviation
perspectives have informed the analysis of asymptotics for spherical integrals
[11, 23], it is possible that a similar approach could inform asymptotics for
integrals over the Stiefel manifold, which arise, for instance, as the
normalizing constant of the matrix Bingham distribution [12], or in the study
of multi-spiked random covariance matrices [24].

######  Remark 2.9.

The first term in the definition (2.10) of $\mathbb{H}_{k}$ is the relative
entropy with respect to the $k$-dimensional standard Gaussian measure, which
is (due to Sanov’s theorem) the large deviation rate function for the sequence
of empirical measures of the rows of an $n\times k$ matrix of i.i.d. standard
Gaussian elements. Hence, Theorem 2.8 offers a way of characterizing the
distinction between Haar-distributed matrices on the Stiefel manifold and
Gaussian matrices. Outside of the large deviations literature, a different
comparison between such Stiefel and Gaussian matrices can be found in [29],
which analyzes expectations of sublinear convex functions of random matrices.

######  Remark 2.10.

The second term in the definition (2.10) of $\mathbb{H}_{k}$ arises from the
orthogonality and normalization constraint defining the Stiefel manifold. Note
that because
$\mathbb{P}(\mathbf{A}_{n,k}^{\intercal}\mathbf{A}_{n,k}=I_{k})=1$, we have,
$\mathbb{P}\text{-a.s.}$,

$\displaystyle
0=\text{tr}\left(I_{k}-\mathbf{A}_{n,k}^{\intercal}\mathbf{A}_{n,k}\right)=\text{tr}(I_{k}-\mathcal{C}(\mathsf{L}_{n,k})),\quad
n\in\mathbb{N}_{k}.$ (2.13)

Nonetheless, the definition of the rate function $\mathbb{H}_{k}$ includes the
trace term $\text{tr}(I_{k}-\mathcal{C}(\nu))$, and $\mathbb{H}_{k}(\nu)$ is
finite even for $\nu\in\mathcal{P}(\mathbb{R}^{k})$ such that
$\text{tr}(I_{k}-\mathcal{C}(\nu))\neq 0$, due to the fact that the statement
of an LDP (Definition 1.1) involves infimization of the rate function
$\mathbb{H}_{k}$ not over a set like
$\mathcal{V}_{k}:=\\{\nu\in\mathcal{P}(\mathbb{R}^{k}):I_{k}=\mathcal{C}(\nu)\\}$,
but rather over its interior and closure (in the space of probability
measures). In particular, the example set $\mathcal{V}_{k}$ is neither open
nor closed with respect to the weak topology. In fact, it is possible to show
from [31] that $\mathcal{V}_{k}$ is neither open nor closed with respect to
_any_ topology for which the sequence of empirical measures
$\\{\mathsf{L}_{n,k}\\}_{n\in\mathbb{N}_{k}}$ satisfies an LDP.

An immediate consequence of Theorem 2.8 is the following:

######  Corollary 2.11.

Fix $k\in\mathbb{N}$. Then for all $q\in(0,2)$, the sequence
$\\{\mathsf{L}_{n,k}\\}_{n\in\mathbb{N}_{k}}$ satisfies the strong law of
large numbers in $\mathcal{P}_{q}(\mathbb{R}^{k})$. That is, almost surely, as
$n\to\infty$, we have $\mathcal{W}_{q}(\mathsf{L}_{n,k},\gamma_{2}^{\otimes
k})\to 0$.

###### Proof.

By Theorem 2.8, the rate function $\mathbb{H}_{k}$ in (2.10) is strictly
convex. Since $\mathbb{H}_{k}(\gamma^{\otimes k})=0$, $\mathbb{H}_{k}$ attains
its unique minimum over $\mathcal{P}_{q}(\mathbb{R}^{k})$ at
$\gamma_{2}^{\otimes k}$. For $\epsilon>0$, due to the LDP for
$\\{\mathsf{L}_{n,k}\\}_{n\in\mathbb{N}_{k}}$ and the uniqueness of the
minimum of $\mathbb{H}_{k}$, there exists $\delta>0$ and $N\in\mathbb{N}_{k}$
such that for $n>N$,
$\mathbb{P}(\mathcal{W}_{q}(\mathsf{L}_{n,k},\gamma_{2}^{\otimes
k})>\epsilon)\leq e^{-n\delta}$, which combined with the Borel-Cantelli Lemma
yields the almost sure convergence of $\mathsf{L}_{n,k}$. ∎

##  3 The Bartlett decomposition and its consequences

We recall the well known result of Bartlett on the QR decomposition of a
matrix with independent standard Gaussian entries, and derive certain
consequences which will be used in the proofs of the main theorem. Throughout,
let $\mathbb{U}_{k}$ denote the space of $k\times k$ upper triangular
matrices, and recall that $\mathbb{V}_{n,k}$ denotes the Stiefel manifold of
$k$-frames in $\mathbb{R}^{n}$.

######  Definition 3.1.

Fix $k\leq n\in\mathbb{N}$. Let $\mathbf{Z}_{n,k}\in\mathbb{R}^{n\times k}$ be
an $n\times k$ matrix with i.i.d. standard Gaussian elements. Let
$\mathbf{Z}_{n,k}=\mathbf{Q}_{n,k}\mathbf{R}_{n,k}$ be the QR decomposition of
$\mathbf{Z}_{n,k}$ as the product of the semi-orthogonal matrix
$\mathbf{Q}_{n,k}\in\mathbb{V}_{n,k}\subset\mathbb{R}^{n\times k}$ and the
upper triangular matrix
$\mathbf{R}_{n,k}\in\mathbb{U}_{k}\subset\mathbb{R}^{k\times k}$.

######  Proposition 3.2 (Bartlett decomposition [4]).

The law of $\mathbf{Q}_{n,k}$ is $\sigma_{n,k}$, the Haar measure on
$\mathbb{V}_{n,k}$. Moreover, the diagonal entries of $\mathbf{R}_{n,k}$
satisfy $\mathbf{R}_{n,k}(i,i)\sim\chi_{n-i+1}$, the chi distribution with
$n-i+1$ degrees of freedom, for $i=1,\cdots,k$.

######  Remark 3.3.

In fact, the matrices $\mathbf{Q}_{n,k}$ and $\mathbf{R}_{n,k}$ of the
Bartlett decomposition are independent, and moreover, the marginal law of the
off-diagonal entries of $\mathbf{R}_{n,k}$ are also explicitly known; however,
we will not need these facts for our analysis. Also, note that when $k=1$, the
Bartlett decomposition corresponds to the classical polar decomposition of the
$n$-dimensional Gaussian measure.

Let $\mathsf{L}^{\mathbf{Z}}_{n,k}$ denote the empirical measure of the _rows_
of $\mathbf{Z}_{n,k}$,

$\mathsf{L}^{\mathbf{Z}}_{n,k}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{\mathbf{Z}_{n,k}(i,\cdot)}\in\mathcal{P}(\mathbb{R}^{k}).$

Then, due to Proposition 3.2, for $\mathbf{A}_{n,k}$ distributed according to
the Haar measure $\sigma_{n,k}$ on $\mathbb{V}_{n,k}$, we have

$\mathbf{A}_{n,k}\buildrel(d)\over{=}\mathbf{Q}_{n,k}=\mathbf{Z}_{n,k}\mathbf{R}_{n,k}^{-1}.$
(3.1)

In the second equality, we use the fact that $\mathbf{R}_{n,k}$ is (almost
surely) invertible, since it is an upper triangular matrix with diagonal
entries that are all (almost surely) positive. Recalling the definition of
$\mathsf{L}_{n,k}$ from (2.12), and using the representation (3.1), we have
for any Borel set $B\subset\mathbb{R}^{k}$,

$\displaystyle\mathsf{L}_{n,k}(B)$
$\displaystyle\buildrel(d)\over{=}\frac{1}{n}\sum_{r=1}^{n}\delta_{\sqrt{n}\mathbf{Z}_{n,k}(r,\cdot)\mathbf{R}_{n,k}^{-1}}(B)=\mathsf{L}^{\mathbf{Z}}_{n,k}\left(B\tfrac{\mathbf{R}_{n,k}}{\sqrt{n}}\right).$
(3.2)

Fortuitously, each element of the matrix $\mathbf{R}_{n,k}$ can be computed as
a function of the _rows_ of the matrix $\mathbf{Z}_{n,k}$, and can be written
as the image of a linear functional of the measure
$\mathsf{L}^{\mathbf{Z}}_{n,k}$. To be more precise, we recall the Gram-
Schmidt process. For a matrix $Z\in\mathbb{R}^{n\times k}$ with columns
$z_{1},\dots,z_{k}\in\mathbb{R}^{n}$, let

$\displaystyle y_{1}$ $\displaystyle:=z_{1}$ $\displaystyle
q_{1}:=\frac{y_{1}}{\|y_{1}\|_{2}};\hskip 64.60918pt$ $\displaystyle y_{i}$
$\displaystyle:=z_{i}-\sum_{h=1}^{i-1}\langle q_{h},z_{i}\rangle q_{h}$
$\displaystyle q_{i}:=\frac{y_{i}}{\|y_{i}\|_{2}},\quad i=2,\dots,k.$

Then, we have the decomposition $Z=QR$ where $Q=(q_{1},\dots,q_{k})$ and

$R=\begin{pmatrix}\langle q_{1},z_{1}\rangle&\langle
q_{1},z_{2}\rangle&\langle q_{1},z_{3}\rangle&\cdots&\langle
q_{1},z_{k}\rangle\\\ 0&\langle q_{2},z_{2}\rangle&\langle
q_{2},z_{3}\rangle&\cdots&\langle q_{2},z_{k}\rangle\\\ 0&0&\langle
q_{3},z_{3}\rangle&\cdots&\langle q_{3},z_{k}\rangle\\\
\vdots&&&\ddots&\vdots\\\ 0&&&&\langle q_{k},z_{k}\rangle\end{pmatrix}.$ (3.3)

Note that we also have the following relation among the elements of $R$: for
$1\leq i\leq j\leq k$,

$\langle q_{i},z_{j}\rangle=\frac{\langle
y_{i},z_{j}\rangle}{\|y_{i}\|_{2}}=\frac{\langle
z_{i},z_{j}\rangle-\sum_{h=1}^{i-1}\langle q_{h},z_{i}\rangle\langle
q_{h},z_{j}\rangle}{\left(\|z_{i}\|^{2}-\sum_{h=1}^{i-1}\langle
q_{h},z_{i}\rangle^{2}\right)^{1/2}}.$ (3.4)

This expression allows us to clarify the relationship between
$\mathbf{R}_{n,k}$ and $\mathsf{L}^{\mathbf{Z}}_{n,k}$.

######  Definition 3.4.

Let $\text{Sym}_{k}$ be the space of real symmetric $k\times k$ matrices. For
$L,M\in\text{Sym}_{k}$, we write $L\succeq M$ (resp., $L\succ M$) if $L-M$ is
positive semi-definite (resp., positive definite).

We equip $\text{Sym}_{k}\subset\mathbb{R}^{k\times k}$ with the induced Borel
$\sigma$-algebra when viewing it as a measurable space, and the Frobenius norm
when viewed as a Banach space.

######  Definition 3.5.

Define the map $\Gamma:\mathbb{R}^{k\times k}\rightarrow\mathbb{U}_{k}$
according to the following iterative procedure: for $M\in\mathbb{R}^{k\times
k}$, $j=1,\dots,k$, and $i=1,\dots,j$,

$\Gamma(M)_{ij}:=\frac{M_{ij}-\sum_{h=1}^{i-1}\Gamma(M)_{hi}\,\Gamma(M)_{hj}}{\left(M_{ii}-\sum_{h=1}^{i-1}\Gamma(M)_{hi}^{2}\right)^{1/2}}\,.$
(3.5)

In the case $i=j=1$, we abide by the convention that empty summations are set
to zero, so $\Gamma(M)_{11}:=M_{11}^{1/2}$.

######  Lemma 3.6.

We have $\mathbb{P}$-a.s.,
$n^{-1/2}\mathbf{R}_{n,k}=\Gamma(\mathcal{C}(\mathsf{L}^{\mathbf{Z}}_{n,k}))$,
where $\mathcal{C}$ is the covariance map of (2.9).

###### Proof.

The result follows from (3.3) and (3.4) upon noticing that for $1\leq i\leq
j\leq k$,

$\displaystyle\mathcal{C}_{ij}(\mathsf{L}^{\mathbf{Z}}_{n,k})$
$\displaystyle=\frac{1}{n}\sum_{r=1}^{n}\mathbf{Z}_{n,k}(r,i)\mathbf{Z}_{n,k}(r,j)=\frac{1}{n}\langle\,\mathbf{Z}_{n,k}(\cdot,i),\mathbf{Z}_{n,k}(\cdot,j)\,\rangle.$

∎

######  Example 3.7.

For example, when $k=1$, we have
$\frac{\mathbf{R}_{n,1}}{\sqrt{n}}=\mathcal{C}_{11}(\mathsf{L}_{n,1})^{1/2}=\frac{\|\mathbf{Z}_{n,1}\|_{2}}{\sqrt{n}}$,
and for $k=2$, we have

$\frac{\mathbf{R}_{n,2}}{\sqrt{n}}=\begin{pmatrix}\mathcal{C}_{11}(\mathsf{L}_{n,2})&\frac{\mathcal{C}_{12}(\mathsf{L}_{n,2})}{\mathcal{C}_{11}(\mathsf{L}_{n,2})^{1/2}}\\\
\
0&\left(\mathcal{C}_{22}(\mathsf{L}_{n,2})-\frac{\mathcal{C}_{21}(\mathsf{L}_{n,2})^{2}}{\mathcal{C}_{11}(\mathsf{L}_{n,2})}\right)^{1/2}\end{pmatrix}.$

######  Remark 3.8.

Note that if $M$ is symmetric positive semi-definite, then $\Gamma(M)$
computes the _Cholesky decomposition_ of $M$, so that
$\Gamma(M)^{\intercal}\Gamma(M)=M$.

##  4 Proof of the empirical measure large deviations

The representation (3.2) and Lemma 3.6 suggest our plan of attack for the
proof of Theorem 2.8: first prove a joint LDP for
$\\{\mathsf{L}^{\mathbf{Z}}_{n,k},\mathcal{C}(\mathsf{L}^{\mathbf{Z}}_{n,k})\\}_{n\in\mathbb{N}_{k}}$;
then, establish an LDP for $\\{\mathsf{L}_{n,k}\\}_{n\in\mathbb{N}_{k}}$. Note
that we do not attempt to directly establish an LDP for
$\\{\mathsf{L}_{n,k}\\}_{n\in\mathbb{N}_{k}}$ from that for
$\\{\mathsf{L}^{\mathbf{Z}}_{n,k}\\}_{n\in\mathbb{N}_{k}}$, because the map
$\mathsf{L}^{\mathbf{Z}}_{n,k}\mapsto\mathsf{L}_{n,k}$ is not continuous with
respect to the weak topology. Nor do we attempt to directly establish an LDP
for
$\\{\mathsf{L}^{\mathbf{Z}}_{n,k},\Gamma(\mathcal{C}(\mathsf{L}^{\mathbf{Z}}_{n,k}))\\}_{n\in\mathbb{N}_{k}}$,
because the map $\Gamma\circ\mathcal{C}$ is a nonlinear functional on the
space of measures on $\mathbb{R}^{k}$. In contrast, our proposed first step is
tractable precisely because $\mathcal{C}$ is a linear functional. We will make
use of the following result, which is stated in [17, Corollary A.2] as a
corollary of [3, Proposition 6.4].

######  Lemma 4.1 (approximate contraction principle).

Let $\Sigma$ be a Polish space and ${\mathcal{X}}$ be a separable Banach space
with topological dual $\mathcal{X}^{*}$. Let
$\\{\mathscr{L}_{n}\\}_{n\in\mathbb{N}}$ be a sequence of
$\mathcal{P}(\Sigma)$-valued random variables such that for each
$n\in\mathbb{N}$, $\mathscr{L}_{n}$ is the empirical measure of $n$ i.i.d.
$\Sigma$-valued random variables $\mathsf{s}_{1},\dots,\mathsf{s}_{n}$ with
common distribution $\mu$ (that does not depend on $n$). For any continuous
$W:\Sigma\rightarrow\mathbb{R}$, define

$\widehat{\Lambda}(W):=\log\mathbb{E}[e^{W(\mathsf{s}_{1})}].$ (4.1)

Also, let $\mathbf{c}:\Sigma\mapsto{\mathcal{X}}$ be a continuous map such
that $0$ lies in the interior $\mathscr{D}^{\circ}$ of the set

$\mathscr{D}:=\left\\{\alpha\in\mathcal{X}^{*}:\widehat{\Lambda}(\langle\alpha,\mathbf{c}(\cdot)\rangle)<\infty\right\\},$
(4.2)

and let $\mathscr{C}_{n}:=\int_{\Sigma}\mathbf{c}(x)\mathscr{L}_{n}(dx)$.
Lastly, define $F:\mathcal{X}\rightarrow\mathbb{R}$ as

$F(x):=\sup_{\alpha\in\mathscr{D}_{\circ}}\langle\alpha,x\rangle,\quad
x\in\mathcal{X}.$ (4.3)

Then, $\\{\mathscr{L}_{n},\mathscr{C}_{n}\\}_{n\in\mathbb{N}}$ satisfies an
LDP with the GRF
$\mathbb{I}:\mathcal{P}(\Sigma)\times\mathcal{X}\rightarrow[0,\infty]$ defined
by

$\mathbb{I}(\nu,x):=\left\\{\begin{array}[]{ll}H(\nu|\mu)+F\left(x-\int_{\Sigma}\mathbf{c}\,d\nu\right)&\textnormal{
if }H(\nu|\mu)<\infty,\\\ +\infty&\textnormal{
else,}\end{array}\right.\quad\nu\in\mathcal{P}(\Sigma),x\in\mathcal{X}.$ (4.4)

######  Lemma 4.2.

For any $k\in\mathbb{N}$, the sequence
$\\{\mathsf{L}^{\mathbf{Z}}_{n,k},\mathcal{C}(\mathsf{L}^{\mathbf{Z}}_{n,k})\\}_{n\in\mathbb{N}_{k}}$
satisfies an LDP in $\mathcal{P}(\mathbb{R}^{k})\times\textnormal{Sym}_{k}$
with GRF
$\mathbb{J}_{k}:\mathcal{P}(\mathbb{R}^{k})\times\textnormal{Sym}_{k}\rightarrow[0,\infty]$,
defined, for $\nu\in\mathcal{P}(\mathbb{R}^{k})$ and
$M\in\textnormal{Sym}_{k}$, to be

$\mathbb{J}_{k}(\nu,M):=\left\\{\begin{array}[]{ll}H(\nu|\gamma^{\otimes
k})+\tfrac{1}{2}\textnormal{tr}\left(M-\int_{\mathbb{R}^{k}}[z\otimes
z]\,\nu(dz)\right)&\textnormal{ if }\int_{\mathbb{R}^{k}}[z\otimes
z]\,\nu(dz)\preceq M,\\\ +\infty&\textnormal{ else.}\end{array}\right.$ (4.5)

###### Proof.

We invoke the approximate contraction principle of Lemma 4.1 with the
following parameters: $\Sigma=\mathbb{R}^{k}$; $\mathcal{X}=\text{Sym}_{k}$
and $\mathcal{X}^{*}=\text{Sym}_{k}$; $\mathbf{c}(z):=[z\otimes z]$ for
$z\in\mathbb{R}^{k}$;
$\mathscr{L}_{n}:=\frac{1}{n}\sum_{j=1}^{n}\delta_{\mathsf{s}_{j}}\mbox{ for
}\mathsf{s}_{1},\mathsf{s}_{2},\dots$ i.i.d. random vectors with common
distribution $\gamma^{\otimes k};$ and
$\mathscr{C}_{n}:=\int_{\mathbb{R}^{k}}\mathbf{c}\,d\mathscr{L}_{n}$. Note
that

$(\mathsf{L}^{\mathbf{Z}}_{n,k},\mathcal{C}(\mathsf{L}^{\mathbf{Z}}_{n,k}))\buildrel(d)\over{=}(\mathscr{L}_{n},\mathscr{C}_{n}),\qquad
n\in\mathbb{N}_{k}.$ (4.6)

With $\widehat{\Lambda}$ as defined in (4.1), the domain $\mathscr{D}$
specified in (4.2) takes the form

$\displaystyle\mathscr{D}$
$\displaystyle=\left\\{\zeta\in\text{Sym}_{k}:\log\int_{\mathbb{R}^{k}}\exp\left(\langle\zeta,\mathbf{c}(z)\rangle\right)\gamma^{\otimes
k}(dz)<\infty\right\\}$
$\displaystyle=\left\\{\zeta\in\text{Sym}_{k}:\log\int_{\mathbb{R}^{k}}\frac{1}{(2\pi)^{k/2}}\exp\left(-z^{\intercal}(\tfrac{1}{2}I_{k}-\zeta)z\right)dz<\infty\right\\}$
$\displaystyle=\left\\{\zeta\in\text{Sym}_{k}:\tfrac{1}{2}I_{k}-\zeta\succ
0\right\\}.$

This last expression indicates that $\mathscr{D}$ is a shifted reflection of
the positive definite cone, hence open, implying that
$\mathscr{D}^{\circ}=\mathscr{D}$. This expression for the form of
$\mathscr{D}$ also makes it clear that $0\in\mathscr{D}^{\circ}$. Lastly the
value of the supremum in the definition of $F$ takes an explicit form due to
the linearity of the trace functional and the fact that the constraint set
$\mathscr{D}$ is a cone (due to the positive definiteness constraint): for
$\eta\in\text{Sym}_{k}$,

$F(\eta)=\sup_{\zeta\in\text{Sym}_{k}}\left\\{\text{Tr}(\zeta\eta):\zeta\prec\tfrac{1}{2}I_{k}\right\\}=\left\\{\begin{array}[]{cl}\frac{1}{2}\text{Tr}(\eta)&\text{
if }\eta\succeq 0,\\\ +\infty&\text{ else.}\end{array}\right.$

Therefore, (4.6) and Lemma 4.1 together imply that
$\\{(\mathsf{L}^{\mathbf{Z}}_{n,k},\mathcal{C}(\mathsf{L}^{\mathbf{Z}}_{n,k}))\\}_{n\in\mathbb{N}_{k}}$
satisfies the stated LDP. ∎

We now establish a relation between the rate function $\mathbb{J}_{k}$ and the
rate function $\mathbb{H}_{k}$ of Theorem 2.8.

######  Lemma 4.3.

For any $k\in\mathbb{N}$ and $\nu\in\mathcal{P}(\mathbb{R}^{k})$, given
$\mathbb{H}_{k}$ of (2.10), $\mathbb{J}_{k}$ of (4.5), and $\Gamma$ of(3.5),
we have

$\mathbb{H}_{k}(\nu)=\widetilde{\mathbb{H}}_{k}(\nu):=\inf_{M\in\textnormal{Sym}_{k}}\mathbb{J}_{k}(\nu(\,\cdot\times\Gamma(M)^{-1}),M).$

###### Proof.

As noted in Remark 3.8, for $M\in\text{Sym}_{k}$ and $\Gamma$ as in (3.5), we
have $M=\Gamma(M)^{\intercal}\Gamma(M)$. Given this equality, the constraint
$M\succeq\int_{\mathbb{R}^{k}}[x\otimes x]\,\nu(dx\times\Gamma(M)^{-1})$ can
be rewritten, using the notation $\mathcal{C}$ from (2.9), as

$I_{k}\succeq\int_{\mathbb{R}^{k}}[x\otimes x]\nu(dx)=\mathcal{C}(\nu).$

If the preceding constraint is satisfied, then using the form of
$\mathbb{J}_{k}$ in (4.5) in the first equality below, the chain rule for
relative entropy in the third equality, and then the form of the Gaussian
distribution $\gamma^{\otimes k}$, we obtain

$\displaystyle\mathbb{J}_{k}$
$\displaystyle(\nu(\,\cdot\times\Gamma(M)^{-1}),M)$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{k}\left(M_{ii}-\int_{\mathbb{R}^{k}}x_{i}^{2}\,\nu(dx\times\Gamma(M)^{-1})\right)+H(\nu(\,\cdot\times\Gamma(M)^{-1})|\gamma^{\otimes
k})$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{k}M_{ii}-\frac{1}{2}\int_{\mathbb{R}^{k}}y^{\intercal}\,\Gamma(M)\Gamma(M)^{\intercal}y\,\nu(dy)+H(\nu|\gamma^{\otimes
k}(\,\cdot\times\Gamma(M)))$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{k}M_{ii}-\frac{1}{2}\int_{\mathbb{R}^{k}}y^{\intercal}\,\Gamma(M)\Gamma(M)^{\intercal}y\,\nu(dy)-\int_{\mathbb{R}^{k}}\log(\tfrac{d\gamma^{\otimes
k}(\,\cdot\times\Gamma(M))}{d\gamma^{\otimes k}})\,d\nu+H(\nu|\gamma^{\otimes
k})$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{k}M_{ii}-\frac{1}{2}\int_{\mathbb{R}^{k}}y^{\intercal}\,\Gamma(M)\Gamma(M)^{\intercal}y\,\nu(dy)+\frac{1}{2}\log\det(\Gamma(M)^{-\intercal}\Gamma(M)^{-1})$
$\displaystyle\quad\quad\quad+\frac{1}{2}\int_{\mathbb{R}^{k}}y^{\intercal}\,(\Gamma(M)\Gamma(M)^{\intercal}-I_{k})y\,\nu(dy)+H(\nu|\gamma^{\otimes
k})$

Due to the upper triangular structure of $\Gamma(M)$, we have

$\frac{1}{2}\log\det(\Gamma(M)^{-\intercal}\Gamma(M)^{-1})=-\log\det(\Gamma(M))=-\sum_{i=1}^{k}\log\Gamma(M)_{ii}.$

Also, note that
$\text{Tr}(I_{k}-\mathcal{C}(\nu))=k-\int_{\mathbb{R}^{k}}y^{\intercal}y\,\nu(dy)$.
Hence, invoking the definition of $\mathbb{H}_{k}$ in (2.10), we have

$\displaystyle\mathbb{J}_{k}(\nu(\,\cdot\times\Gamma(M)^{-1}),M)$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{k}(M_{ii}-1)-\sum_{i=1}^{k}\log\Gamma(M)_{ii}+\mathbb{H}_{k}(\nu).$

Taking the infimum of the expression above over $M\in\textnormal{Sym}_{k}$, we
see that

$\displaystyle\widetilde{\mathbb{H}}_{k}(\nu)$
$\displaystyle=\inf_{M\in\textnormal{Sym}_{k}}\left\\{\sum_{i=1}^{k}\left(\frac{M_{ii}-1}{2}-\log\Gamma(M)_{ii}\right)\right\\}+\mathbb{H}_{k}(\nu).$
(4.7)

Note that by the definition of $\Gamma$ in (3.5),

$\Gamma(M)_{ii}=\left(M_{ii}-\sum_{h=1}^{i-1}\Gamma(M)_{hi}^{2}\right)^{1/2},$

and by the definition of the Gram-Schmidt process, we have
$M_{ii}\geq\sum_{h=1}^{i-1}\Gamma(M)_{hi}^{2}\geq 0$. Thus, for all
$i=1,\dots,k$, for any fixed $M_{ii}$, the maximum value of $\Gamma(M)_{ii}$
is attained when $\Gamma(M)_{hi}=0$ for $h=1,\dots,i-1$. Therefore, once again
using $M=\Gamma(M)^{\intercal}\Gamma(M)$, we obtain

$\displaystyle\inf_{M\in\textnormal{Sym}_{k}}\left\\{\sum_{i=1}^{k}\left(\frac{M_{ii}-1}{2}-\log\Gamma(M)_{ii}\right)\right\\}$
$\displaystyle=\inf_{M_{ii}\geq
0,i=1,\dots,k}\left\\{\frac{1}{2}\sum_{i=1}^{k}\left(M_{ii}-1-\log
M_{ii}\right)\right\\}$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{k}\inf_{M_{ii}\geq
0}\left\\{M_{ii}-1-\log M_{ii}\right\\},$

which is clearly equal to zero. Together with (4.7), this shows that
$\mathbb{H}_{k}=\widetilde{\mathbb{H}}_{k}$. ∎

######  Lemma 4.4.

Fix $k\in\mathbb{N}$ and consider the following set of probability measures,

$\mathcal{K}:=\left\\{\nu\in\mathcal{P}(\mathbb{R}^{k}):\int_{\mathbb{R}^{k}}\|x\|^{2}\nu(dx)\leq
k\right\\}.$ (4.8)

For any $q\in(0,2)$, the set
$\mathcal{K}\subset\mathcal{P}_{2}(\mathbb{R}^{d})$ is compact with respect to
the $q$-Wasserstein topology. In addition, $\mathcal{K}$ is convex and non-
empty.

###### Proof.

The proof is an elementary modification of the proof of the $k=1$ case given
in [20, Lemma 3.14]. ∎

###### Proof of Theorem 2.8.

Let $\Gamma$ be as defined in (3.5). Due to (3.2) and Lemma 3.6, we have

$\mathsf{L}_{n,k}\buildrel(d)\over{=}\mathsf{L}^{\mathbf{Z}}_{n,k}(\,\cdot\times\tfrac{\mathbf{R}_{n,k}}{\sqrt{n}})=\mathsf{L}^{\mathbf{Z}}_{n,k}\left(\,\cdot\times\Gamma\left(\mathcal{C}(\mathsf{L}^{\mathbf{Z}}_{n,k})\right)\right).$

The image of $\mathcal{C}$ is positive semi-definite matrices, so as noted in
Remark 3.8, the map $\Gamma$ maps a matrix to its Cholesky decomposition,
hence $M\mapsto\Gamma(M)$ is continuous. By Slutsky’s theorem, the map

$\mathcal{P}(\mathbb{R}^{k})\times\textnormal{Sym}_{k}\ni(\mu,M)\mapsto\mu(\,\cdot\,\times\Gamma(M))\in\mathcal{P}(\mathbb{R}^{k}),$

is also continuous. An application of the contraction principle to the map
above yields an LDP for the sequence
$\\{\mathsf{L}_{n,k}\\}_{n\in\mathbb{N}_{k}}$ in $\mathcal{P}(\mathbb{R}^{k})$
(i.e., with respect to the weak topology), with GRF

$\displaystyle\inf_{\nu\in\mathcal{P}(\mathbb{R}^{k}),M\in\textnormal{Sym}_{k}}\\{\mathbb{J}_{k}(\mu,M):\nu=\mu(\,\cdot\times\Gamma(M))\\}=\inf_{M\in\textnormal{Sym}_{k}}\mathbb{J}_{k}(\nu(\,\cdot\times\Gamma(M)^{-1}),M)=\mathbb{H}_{k}(\nu),$

where the last equality is due to Lemma 4.3.

Fix $q\in(0,2)$. In order to establish the LDP for
$\\{\mathsf{L}_{n,k}\\}_{n\in\mathbb{N}_{k}}$ in
$\mathcal{P}_{q}(\mathbb{R}^{k})$ (i.e., with respect to the stronger
$q$-Wasserstein topology), by Corollary 4.2.6 of [8], it suffices to show
exponential tightness of $\\{\mathsf{L}_{n,k}\\}_{n\in\mathbb{N}_{k}}$ in the
$q$-Wasserstein topology. Let $\mathcal{K}$ be the set defined in (4.8), which
is compact (with respect to the $q$-Wasserstein topology) due to Lemma 4.4.
Note that $\int_{\mathbb{R}^{k}}|x|^{2}\mathsf{L}_{n,k}(dx)=k$ a.s., so
$\mathbb{P}(\mathsf{L}_{n,k}\in\mathcal{K}_{2,k}^{c})=0$, and hence,
$\log\mathbb{P}(\mathsf{L}_{n,k}\in\mathcal{K}_{2,k}^{c})=-\infty$ for all
$n\in\mathbb{N}_{k}$, trivially implying the exponential tightness of
$\\{\mathsf{L}_{n,k}\\}_{n\in\mathbb{N}_{k}}$.

Lastly, the strict convexity of $\mathbb{H}_{k}$ follows from the strict
convexity of the relative entropy $H(\cdot|\gamma^{\otimes k})$ and the
linearity of the covariance map $\mathcal{C}$. ∎

##  5 Proof of the quenched LDP

In this section, we state the proof of Theorem 2.4. As a precursor, we state
two lemmas that will assist with part (iii) of the theorem.

######  Lemma 5.1.

Fix $m\in\mathbb{N}$, and let $\mathscr{F}$ be a set of functions from
$\mathbb{R}^{m}$ to $\mathbb{R}$ such that every $f\in\mathscr{F}$ is
symmetric about 0 and convex. Then, defining
$g:\mathbb{R}^{m}\rightarrow\mathbb{R}$ as

$g(x):=\inf_{f\in\mathscr{F}}f(x),\quad x\in\mathbb{R}^{m},$

the function $g$ is monotone with respect to scaling in the sense that for all
$x\in\mathbb{R}^{m}$, the mapping

$\mathbb{R}_{+}\ni c\mapsto g(cx)\in\mathbb{R}$ (5.1)

is non-decreasing.

###### Proof.

Fix $x\in\mathbb{R}^{m}$ and $c_{1}<c_{2}\in\mathbb{R}_{+}$. For any
$f\in\mathscr{F}$, the symmetry about 0 and convexity of $f$ implies that $f$
has a global minimum at 0, hence

$\displaystyle f(c_{1}x)$ $\displaystyle=f(\tfrac{c_{1}}{c_{2}}\times
c_{2}x+\tfrac{c_{2}-c_{1}}{c_{2}}\times 0)$
$\displaystyle\leq\tfrac{c_{1}}{c_{2}}f(c_{2}x)+\tfrac{c_{2}-c_{1}}{c_{2}}f(0)$
$\displaystyle=f(c_{2}x)+\tfrac{c_{2}-c_{1}}{c_{2}}(f(0)-f(c_{2}x))$
$\displaystyle\leq f(c_{2}x),$

where the first inequality follows from convexity, and the second inequality
is due to the global minimum at 0. Taking the infimum over all
$f\in\mathscr{F}$ on both sides, we find that $g(c_{1}x)\leq g(c_{2}x)$,
completing the proof. ∎

######  Lemma 5.2.

Fix $m\in\mathbb{N}$, and let
$\mathsf{x}=\\{\mathsf{x}_{n}\\}_{n\in\mathbb{N}}$ denote a sequence of
$\mathbb{R}^{m}$-valued random variables that satisfies an LDP with GRF
$I_{\mathsf{x}}$. Let $U$ be a uniformly distributed random variable on
$[0,1]$ independent of $\\{\mathsf{x}_{n}\\}_{n\in\mathbb{N}}$. If for all
$x\in\mathbb{R}^{m}$, the mapping $\mathbb{R}_{+}\ni c\mapsto
I_{\mathsf{x}}(cx)\in[0,\infty]$ is non-decreasing, then the scaled sequence
$\\{U^{1/n}\mathsf{x}_{n}\\}_{n\in\mathbb{N}}$ satisfies an LDP with GRF
$I_{\mathsf{x}}$.

###### Proof.

Due to [10, Lemma 3.3], the sequence $\\{U^{1/n}\\}_{n\in\mathbb{N}}$
satisfies an LDP with the good rate function

$I_{U}(u):=\left\\{\begin{array}[]{ll}-\log u&u\in(0,1];\\\
+\infty&\textnormal{ else. }\end{array}\right.$

By independence, the sequence
$\\{U^{1/n},\,\,\mathsf{x}_{n}\\}_{n\in\mathbb{N}}$ satisfies a joint LDP with
the GRF $I_{U,\mathsf{x}}:\mathbb{R}\times\mathbb{R}^{m}$ defined as
$I_{U,\mathsf{x}}(u,w):=I_{U}(u)+I_{\mathsf{x}}(x)$. By the contraction
principle, the scaled sequence $\\{U^{1/n}\mathsf{x}_{n}\\}_{n\in\mathbb{N}}$
satisfies an LDP with the rate function $I$, where for $x\in\mathbb{R}^{m}$,

$\displaystyle I(x)$
$\displaystyle:=\inf_{u\in\mathbb{R},\,y\in\mathbb{R}^{m}}\\{I_{U}(u)+I_{\mathsf{x}}(y):uy=x\\}=\inf_{u\in(0,1]}\\{-\log
u+I_{\mathsf{x}}(\tfrac{x}{u})\\}.$

The mapping $u\mapsto 1/u$ is monotonically decreasing, which when combined
with the assumption on $I_{\mathsf{x}}$ implies that $u\mapsto
I_{\mathsf{x}}(x/u)$ is monotonically decreasing. Since $u\mapsto-\log u$ is
also monotonically decreasing, the infimum above is attained at $u=1$, hence
$I(x)=I_{\mathsf{x}}(x)$ for all $x\in\mathbb{R}^{m}$. ∎

###### Proof of Theorem 2.4.

Suppose Assumption 2.1 holds for some $\\{\xi_{j}\\}_{j\in\mathbb{N}}$,
$\mathbf{r}$, $\rho$, $q_{\star}>0$, and $T\leq\infty$, all as defined in the
statement of the assumption. We first prove an LDP for the following
$\mathbb{R}^{k+1}$-valued sequence:

$R_{\mathbf{a}}^{(n)}:=\left(n^{-1/2}\,\mathbf{a}_{n,k}^{\intercal}\xi^{(n)},\,\,\frac{1}{n}\sum_{i=1}^{n}\mathbf{r}(\xi_{i})\right),\qquad
n\in\mathbb{N}.$ (5.2)

In terms of the log mgf $\Lambda$ of $(\xi_{1},\mathbf{r}(\xi_{1}))$, the
scaled log mgf of $R_{\mathbf{a}}^{(n)}$ takes the form: for
$t_{1}\in\mathbb{R}^{k}$ and $t_{2}\in\mathbb{R}$,

$\displaystyle\frac{1}{n}\log\mathbb{E}\left[\exp(n\,\langle
t,R_{\mathbf{a}}^{(n)}\rangle)\right]$
$\displaystyle=\frac{1}{n}\log\mathbb{E}\left[\exp\left(\sum_{i=1}^{n}\left(\sqrt{n}\,\xi_{i}\langle
t_{1},\mathbf{a}_{n,k}(i,\cdot)\rangle+t_{2}\,\mathbf{r}(\xi_{i})\right)\right)\right]$
$\displaystyle=\frac{1}{n}\log\prod_{i=1}^{n}\mathbb{E}\left[\exp\left(\sqrt{n}\,\xi_{i}\langle
t_{1},\mathbf{a}_{n,k}(i,\cdot)\rangle+t_{2}\,\mathbf{r}(\xi_{i})\right)\right]$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\Lambda\left(\,\langle
t_{1},\sqrt{n}\mathbf{a}_{n,k}(i,\cdot)\rangle,t_{2}\right)$
$\displaystyle=\Psi_{\mathsf{L}_{n,k}^{\mathbf{a}}}(t_{1},t_{2}),$

with $\Psi_{\cdot}$ equal to the integrated log mgf functional defined in
(2.3).

Fix $t_{1}\in\mathbb{R}^{k}$. For $t_{2}\geq T$, both sides equal $+\infty$
due to Remark 2.2. For $t_{2}<T$, due to the $q_{\star}$-Wasserstein
continuity pointed out in Remark 2.2, together with the
$q_{\star}$-Wasserstein convergence of (2.6), we take the limit as
$n\rightarrow\infty$ to find

$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{E}\left[\exp(n\,\langle
t,R_{\mathbf{a}}^{(n)}\rangle)\right]$
$\displaystyle=\Psi_{\nu}(t_{1},t_{2}).$

Due to the lower semicontinuity and essential smoothness of $\Psi_{\nu}$,
which follow from Assumption 2.1(iii), the Gärtner-Ellis theorem (see, e.g.,
[8, Theorem 2.3.6]) yields the LDP for the sequence
$\\{R_{\mathbf{a}}^{(n)}\\}_{n\in\mathbb{N}}$ in $\mathbb{R}^{k+1}$ with the
GRF $\Psi_{\nu}^{*}$ from (2.5).

Due to the representation of $X^{(n)}$ given by Assumption 2.1(i), we have

$n^{-1/2}\mathbf{A}_{n,k}^{\intercal}X^{(n)}\buildrel(d)\over{=}\rho\left((R_{\mathbf{a}}^{(n)})_{2}\right)(R_{\mathbf{a}}^{(n)})_{1},$

where $\rho$ is continuous. The LDP for
$\\{R_{\mathbf{a}}^{(n)}\\}_{n\in\mathbb{N}}$ and the contraction principle
applied to the continuous mapping
$\mathbb{R}^{k+1}\ni(\tau_{1},\tau_{2})\mapsto\rho(\tau_{2})\tau_{1}\in\mathbb{R}^{k}$
yield an LDP for
$\\{n^{-1/2}\mathbf{a}_{n,k}^{\intercal}X^{(n)}\\}_{n\in\mathbb{N}_{k}}$ in
$\mathbb{R}^{k}$ with good rate function $\bar{\mathcal{J}}_{\nu}^{\sf{qu}}$
defined to be

$\bar{\mathcal{J}}_{\nu}^{\sf{qu}}(x):=\inf_{\tau_{1}\in\mathbb{R}^{k},\,\tau_{2}\in\mathbb{R}_{+}}\left\\{\Psi_{\nu}^{*}(\tau_{1},\tau_{2}):\tau_{1}\rho(\tau_{2})=x\right\\},\quad
x\in\mathbb{R}^{k}.$

The equivalence of $\bar{\mathcal{J}}_{\nu}^{\sf{qu}}$ to the rate function
$\mathcal{J}_{\nu}^{\sf{qu}}$ in (2.7) follows from using the constraint
$\tau_{1}\rho(\tau_{2})=x$ to substitute for the first argument in
$\Psi_{\nu}^{*}$. This proves part (i) of the theorem.

In turn, the LDP from part (i) implies part (ii) of the theorem since by
Corollary 2.11, almost surely,
${\mathcal{W}}_{q_{\star}}(\mathsf{L}_{n,k}^{\mathbf{A}},\gamma^{\otimes
k})={\mathcal{W}}_{q_{\star}}(\mathsf{L}_{n,k},\gamma^{\otimes k})\rightarrow
0$ as $n\rightarrow\infty$.

We turn to the final claim (iii). Given the assumption on symmetry of
$\Lambda$, it is apparent from the definition (2.3) that $\Psi_{\nu}$ is
symmetric in its first argument, and then from the definition of the Legendre
transform (2.5) that $\Psi_{\nu}^{*}$ is also symmetric in its first argument.
Applying Lemma 5.1 with dimension $m=k$, the set of symmetric convex functions
$\mathscr{F}=\\{\mathbb{R}^{k}\ni
x\mapsto\Psi_{\nu}^{*}(\frac{x}{\rho(\tau)},\tau)\in\mathbb{R}\\}_{\tau\in\mathbb{R}_{+}}$,
and $g=\mathcal{J}_{\nu}^{\sf{qu}}$, we find that the mapping
$\mathbb{R}_{+}\ni c\mapsto\mathcal{J}_{\nu}^{\sf{qu}}(cx)\in[0,\infty]$ is
non-decreasing. An application of Lemma 5.2 with
$\mathsf{x}_{n}=n^{-1/2}\mathbf{a}_{n,k}^{\intercal}X^{(n)}$ for
$n\in\mathbb{N}$ and $I_{\mathsf{x}}=\mathcal{J}_{\nu}^{\sf{qu}}$ completes
the proof. ∎

##  6 Proof of the variational formula

In this section, we prove Theorem 2.7, primarily through an application of
Theorem 2.8 and Sion’s minimax theorem [27]. We start with preliminary results
in Lemmas 6.1, 6.2, and 6.3. Throughout, recall the definition of
$\mathbb{H}_{k}$ from (2.10).

######  Lemma 6.1.

Suppose Assumption 2.1 holds, with associated $T$ and $\Psi_{\nu}$,
$\nu\in\mathcal{P}(\mathbb{R}^{k})$, and recall the empirical measure
$\mathsf{L}_{n,k}$ from (2.12). For $t_{1}\in\mathbb{R}^{k}$, $t_{2}<T$, and
$0<\delta<\infty$, the following condition holds:

$\limsup_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{E}\left[e^{\delta
n\Psi_{\mathsf{L}_{n,k}}(t_{1},t_{2})}\right]<\infty\,.$ (6.1)

###### Proof.

Let $\Theta^{(n)}:=(\Theta_{1}^{(n)},\dots,\Theta_{n}^{(n)})$ denote a random
vector distributed uniformly on the Euclidean sphere in $\mathbb{R}^{n}$ of
radius 1. For $t_{1}\in\mathbb{R}^{k}$, the random vector
$\mathbf{A}_{n,k}t_{1}$ lies on the Euclidean sphere in $\mathbb{R}^{n}$ of
radius $\|t_{1}\|_{2}$ and has a law invariant to orthogonal transformation
(due to the law of $\mathbf{A}_{n,k}$); hence,
$\mathbf{A}_{n,k}t_{1}\buildrel(d)\over{=}\|t_{1}\|_{2}\Theta^{(n)}$. Fix
$t_{1}\in\mathbb{R}^{k}$ and $t_{2}<T$, let $C_{t_{2}}$ and $q_{\star}$ be as
in Assumption 2.1(ii), and for $i\in\mathbb{N}$, define
$g_{i}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ as

$g_{i}(x)=\exp\left(\delta
C_{t_{2}}[1+(\sqrt{n}\|t_{1}\|_{2}x)^{q_{\star}}]\right),\quad
x\in\mathbb{R}_{+}.$

When combined, the relation
$\mathbf{A}_{n,k}t_{1}\buildrel(d)\over{=}\|t_{1}\|_{2}\Theta^{(n)}$, the
bound of Assumption 2.1(ii) and the sub-independence of
$(|\Theta_{1}^{(n)}|,\dots,|\Theta_{n}^{(n)}|)$ as given by [5, Theorem
2.11(2)] with $p=2$ therein, yield

$\mathbb{E}\left[e^{\delta
n\Psi_{\mathsf{L}_{n,k}}(t_{1},t_{2})}\right]=\mathbb{E}\left[\prod_{i=1}^{n}\exp\left(\delta\Lambda(\sqrt{n}\|t_{1}\|_{2}\Theta_{i}^{(n)},t_{2})\right)\right]\leq\mathbb{E}\left[\prod_{i=1}^{n}g_{i}(|\Theta_{i}^{(n)}|)\right]\leq\prod_{i=1}^{n}\mathbb{E}\left[g_{i}(|\Theta_{i}^{(n)}|)\right].$
(6.2)

Now, let $Z^{(n)}:=(Z_{1},\dots,Z_{n})$, and note that for each
$i=1,\ldots,n$,
$\sqrt{n}\Theta_{i}^{(n)}\buildrel(d)\over{=}\sqrt{n}Z_{1}/\|Z^{(n)}\|_{2}\xrightarrow{\textnormal{a.s.}}Z_{1}$.
Therefore, taking the limit superior, as $n\to\infty$, in (6.2) and applying
the reverse Fatou lemma, we obtain

$\displaystyle\limsup_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{E}\left[e^{\delta
n\Psi_{\mathsf{L}_{n,k}}(t_{1},t_{2})}\right]$
$\displaystyle\leq\limsup_{n\rightarrow\infty}\log\mathbb{E}\left[\exp\left(\delta
C_{t_{2}}[1+(\|t_{1}\|_{2}\tfrac{\sqrt{n}|Z_{1}|}{\|Z^{(n)}\|_{2}})^{q_{\star}}]\right)\right]$
$\displaystyle\leq\delta C_{t_{2}}+\log\mathbb{E}\left[\exp(\delta
C_{t_{2}}\|t_{1}\|_{2}^{q_{\star}}\,|Z_{1}|^{q_{\star}})\right].$

Since the last term on the right-hand side is finite for all
$t_{1}\in\mathbb{R}^{k}$ because $q_{\star}<2$, (6.1) follows. ∎

######  Lemma 6.2.

Suppose Assumption 2.1 holds, with associated
$\\{\xi_{j}\\}_{j\in\mathbb{N}}$, $\mathbf{r}$, $T$, and $\Psi_{\nu}$,
$\nu\in\mathcal{P}(\mathbb{R}^{k})$, and let $\mathbf{A}_{n,k}$ be drawn from
the Haar measure $\sigma_{n,k}$ on $\mathbb{V}_{n,k}$, independently of
$\\{\xi_{j}\\}_{j\in\mathbb{N}}$. For $n\in\mathbb{N}$, define

$\Phi_{n}(t_{1},t_{2}):=\frac{1}{n}\log\mathbb{E}\left[\exp\left(\sqrt{n}\,\xi^{(n)}\mathbf{A}_{n,k}t_{1}+t_{2}\sum_{i=1}^{n}\mathbf{r}(\xi_{i})\right)\right],\quad
t_{1}\in\mathbb{R}^{k},\,t_{2}\in\mathbb{R},$ (6.3)

where $\xi^{(n)}:=(\xi_{1},\xi_{2},\ldots,\xi_{n})$. Then, for
$t_{1}\in\mathbb{R}^{k}$ and $t_{2}\in\mathbb{R}$,

$\lim_{n\rightarrow\infty}\Phi_{n}(t_{1},t_{2})=\Phi(t_{1},t_{2}),$

where, with $\mathcal{K}$ equal to the set defined in (4.8),

$\Phi(t_{1},t_{2}):=\sup_{\nu\in\mathcal{P}(\mathbb{R}^{k})}\left\\{\Psi_{\nu}(t_{1},t_{2})-\mathbb{H}_{k}(\nu)\right\\}=\sup_{\nu\in\mathcal{K}}\left\\{\Psi_{\nu}(t_{1},t_{2})-\mathbb{H}_{k}(\nu)\right\\}.$
(6.4)

###### Proof.

Due to the independence of $\xi_{1},\xi_{2},\dots$, and their independence
from $\mathbf{A}_{n,k}$, we can write, for $n\in\mathbb{N}$,

$\displaystyle\Phi_{n}(t_{1},t_{2})$
$\displaystyle=\frac{1}{n}\log\mathbb{E}\left[\prod_{i=1}^{n}\mathbb{E}\left[\exp\left(\sqrt{n}\,\xi_{i}\left(\mathbf{A}_{n,k}t_{1}\right)_{i}+t_{2}\mathbf{r}(\xi_{i})\right)\big{|}\mathbf{A}_{n,k}\right]\right]$
$\displaystyle=\frac{1}{n}\log\mathbb{E}\left[\exp\left(\sum_{i=1}^{n}\Lambda(\sqrt{n}\,\langle\mathbf{A}_{n,k}(i,\cdot),t_{1}\rangle,\,t_{2}\right)\right]$
$\displaystyle=\frac{1}{n}\log\mathbb{E}\left[\exp\left(n\,\Psi_{\mathsf{L}_{n,k}}(t_{1},t_{2})\right)\right],$

for $t_{1}\in\mathbb{R}^{k}$ and $t_{2}\in\mathbb{R}$, where
$\mathsf{L}_{n,k}$ is as in (2.12). Now, let $T\leq\infty$ and
$q_{\star}\in(0,2)$ be as specified in Assumption 2.1(ii). For $t_{2}\geq T$,
by Remark 2.2, both $\Phi_{n}(\cdot,t_{2})$ and $\Phi(\cdot,t_{2})$ are
identically equal to infinity, and so the limit holds trivially. Now, suppose
$t_{2}<T$. Recall from Theorem 2.8 that the sequence
$\\{\mathsf{L}_{n,k}\\}_{n\in\mathbb{N}_{k}}$ satisfies an LDP in
$\mathcal{P}_{q}(\mathbb{R}^{k})$ for all $q\in(0,2)$, with the GRF
$\mathbb{H}_{k}$. Due to the bound in Assumption 2.1(ii), the map
$\mathcal{P}(\mathbb{R})\ni\nu\mapsto\Psi_{\nu}(t_{1},t_{2})\in\mathbb{R}$ is
continuous with respect to the $q_{\star}$-Wasserstein topology, and we know
$q_{\star}\in(0,2)$ due to Assumption 2.1(v). By Varadhan’s lemma [8, Theorem
4.3.1] which is applicable due to Lemma 6.1, it follows that the limit of
$\Phi_{n}(t_{1},t_{2})$ is given by $\Phi(t_{1},t_{2})$ defined in (6.4).

To complete the proof of the lemma, it only remains to establish the last
equality in (6.4), but this is an immediate consequence of the fact that
$\mathbb{H}_{k}(\nu)=\infty$ for $\nu\notin\mathcal{K}$.

∎

######  Lemma 6.3.

Suppose Assumption 2.1 holds, and for each
$\nu\in\mathcal{P}(\mathbb{R}^{k})$, let $\Psi_{\nu}$ be as defined in (2.3),
let $\Psi_{\nu}^{*}$ denote its Legendre transform, as specified in (2.5), and
let $\mathcal{K}\subset\mathcal{P}(\mathbb{R}^{k})$ be the set defined in
(4.8). Then the Legendre transform $\Phi^{*}$ of the function $\Phi$ defined
in (6.4) satisfies, for $\tau_{1}\in\mathbb{R}^{k}$, and
$\tau_{2}\in\mathbb{R}$,

${\Phi}^{*}(\tau_{1},\tau_{2})=\inf_{\nu\in\mathcal{K}}\left\\{\Psi_{\nu}^{*}(\tau_{1},\tau_{2})+\mathbb{H}_{k}(\nu)\right\\}=\inf_{\nu\in\mathcal{P}(\mathbb{R}^{k})}\left\\{\Psi_{\nu}^{*}(\tau_{1},\tau_{2})+\mathbb{H}_{k}(\nu)\right\\}.$
(6.5)

###### Proof.

First, note that the second equality in (6.5) holds because
$\mathbb{H}_{k}(\nu)=\infty$ for $\nu\not\in\mathcal{K}$. Next, fix the
following:

  * •

let $\Lambda$, $T$ be as in Assumption 2.1(ii), and define
$\mathcal{D}_{T}:=\mathbb{R}^{k}\times(-\infty,T)$;

  * •

let $q_{\star}\in(0,2)$ be as in Assumption 2.1(ii), and let
$\mathcal{M}_{q_{\star}}(\mathbb{R}^{k})$ denote the space of finite signed
measures (not necessarily probability measures) on $\mathbb{R}^{k}$, equipped
with the $q_{\star}$-Wasserstein topology.

Fix $\tau=(\tau_{1},\tau_{2})\in\mathbb{R}^{k}\times\mathbb{R}$. Then by the
definition (2.5) of $\Psi_{\nu}^{*}$,

$\displaystyle\Psi_{\nu}^{*}(\tau_{1},\tau_{2})$
$\displaystyle=\sup_{(t_{1},t_{2})\in\mathbb{R}^{k}\times\mathbb{R}}\\{\langle\tau_{1},t_{1}\rangle+\tau_{2}t_{2}-\Psi_{\nu}(t_{1},t_{2})\\}$
$\displaystyle=\sup_{(t_{1},t_{2})\in\mathcal{D}_{T}}\\{\langle\tau_{1},t_{1}\rangle+\tau_{2}t_{2}-\Psi_{\nu}(t_{1},t_{2})\\},$
(6.6)

where the second equality holds because, by Remark 2.2,
$\Psi_{\nu}(t_{1},t_{2})=\infty$ if $t_{2}\geq T$. Thus, the right-hand side
of (6.5) is equal to
$\inf_{\nu\in\mathcal{K}}\sup_{t=(t_{1},t_{2})\in\mathcal{D}_{T}}\inf_{\nu\in\mathcal{K}}F_{\tau}(\nu,t),$
where

$F_{\tau}(\nu,t):=\langle\tau_{1},t_{1}\rangle+\tau_{2}t_{2}-\Psi_{\nu}(t_{1},t_{2})+\mathbb{H}_{k}(\nu),\quad\nu\in\mathcal{P}(\mathbb{R}^{k}),\,t=(t_{1},t_{2})\in\mathbb{R}^{k}\times\mathbb{R}.$

On the other hand, by the definition of $\Phi^{*}$ and the representation
(6.4) for $\Phi$,

$\displaystyle\Phi^{*}(\tau_{1},\tau_{2})$ $\displaystyle=$
$\displaystyle\sup_{(t_{1},t_{2})\in\mathbb{R}^{k+1}}\\{\langle\tau_{1},t_{1}\rangle+\tau_{2}t_{2}-\Phi(t_{1},t_{2})\\}$
$\displaystyle=$
$\displaystyle\sup_{t=(t_{1},t_{2})\in\mathbb{R}^{k+1}}\inf_{\nu\in\mathcal{K}}F_{\tau}(\nu,t),$
$\displaystyle=$
$\displaystyle\sup_{t=(t_{1},t_{2})\in\mathcal{D}_{T}}\inf_{\nu\in\mathcal{K}}F_{\tau}(\nu,t),$

where the last equality uses the fact that for $t_{2}>T$,
$\Psi_{\nu}(t_{1},t_{2})=\infty$ and hence, $F_{\tau}(\nu,t)=-\infty$ (see
Remark 2.2). Thus, to prove the first equality in (6.5), it suffices to show
that for all $(\tau_{1},\tau_{2})\in\mathbb{R}^{k}\times\mathbb{R}$,

$\inf_{\nu\in\mathcal{K}}\sup_{(t_{1},t_{2})\in\mathcal{D}_{T}}F_{\tau}(\nu,(t_{1},t_{2}))=\sup_{(t_{1},t_{2})\in\mathcal{D}_{T}}\inf_{\nu\in\mathcal{K}}F_{\tau}(\nu,(t_{1},t_{2})).$
(6.7)

To justify the exchange of infimum and supremum in (6.7), we verify the
conditions of the minimax theorem [27, Corollary 3.3]. That is, for
$(\tau_{1},\tau_{2})\in\mathbb{R}^{k}\times\mathbb{R}$, we note that

  * •

the set $\mathcal{D}_{T}=\mathbb{R}^{k}\times(-\infty,T)$ is a convex subset
of the topological vector space $\mathbb{R}^{k+1}$;

  * •

due to Lemma 4.4, $\mathcal{K}$ is a convex compact subset of the topological
vector space $\mathcal{M}_{q_{\star}}(\mathbb{R}^{k})$;

  * •

for $t=(t_{1},t_{2})\in\mathcal{D}_{T}$: the lower semicontinuity of
$F_{\tau}(\cdot,t)$ follows from the lower semicontinuity of
$\nu\to\Psi_{\nu}(t)$ due to Assumption 2.1(iii) and of $\mathbb{H}_{k}$ (as
it is a GRF); the convexity of $F_{\tau}(\cdot,t)$ follows from the linearity
of $\nu\mapsto\Psi_{\nu}(t)$ and the convexity of $\mathbb{H}_{k}$, which was
established in Theorem 2.8;

  * •

for $\nu\in\mathcal{K}$: the lower semicontinuity of $t\to\Psi_{\nu}(t)$ on
$\mathcal{D}_{T}$ follows from Assumption 2.1(iii); the convexity of
$\Psi_{\nu}$ on $\mathcal{D}_{T}$ follows from linearity of expectation, the
definition (2.3), and the fact that $\Lambda$ is convex since it is a log mgf;

  * •

since $t\mapsto\langle\tau_{1},t_{1}\rangle+\tau_{2}t_{2}$ is continuous and
linear, it follows that $F_{\tau}(\nu,\cdot)$ is upper semicontinuous and
concave on $\mathcal{D}_{T}$.

Due to the conditions verified above, the minimax theorem can be applied to
conclude that (6.7), and hence, the desired first equality in (6.5), holds.
This completes the proof of the lemma. ∎

###### Proof of Theorem 2.7.

Let $\\{\xi_{j}\\}_{j\in\mathbb{N}}$ and $\mathbf{r}$ be as in Assumption 2.1,
let $\xi^{(n)}:=(\xi_{1},\ldots,\xi_{n})$ and define the sequence
$\\{R^{(n)}\\}_{n\in\mathbb{N}}$ in $\mathbb{R}^{k+1}$ by

$R^{(n)}:=\left(n^{-1/2}\,\mathbf{A}_{n,k}^{\intercal}\xi^{(n)},\,\,\frac{1}{n}\sum_{i=1}^{n}\mathbf{r}(\xi_{i})\right),\qquad
n\in\mathbb{N}.$ (6.8)

First note that the law of $\mathbf{A}_{n,k}$ is invariant to orthogonal
transformation and independent of $\xi^{(n)}$, hence
$\mathbf{A}_{n,k}^{\intercal}\tfrac{\xi^{(n)}}{\|\xi^{(n)}\|_{2}}\buildrel(d)\over{=}\mathbf{A}_{n,k}(1,\cdot)$
and $\mathbf{A}_{n,k}^{\intercal}\tfrac{\xi^{(n)}}{\|\xi^{(n)}\|_{2}}$ is
independent of $\xi^{(n)}$; we refer to [10, Lemma 6.3] for the proof of the
simpler case when $k=1$. As a consequence,

$R^{(n)}\buildrel(d)\over{=}\left(\frac{1}{n}\sqrt{n}\mathbf{A}_{n,k}(1,\cdot)\|\xi^{(n)}\|_{2},\frac{1}{n}\sum_{i=1}^{n}\mathbf{r}(\xi_{i})\right).$

Define the $\mathbb{R}^{k+2}$-valued sequence of random variables,

$S^{(n)}:=\left(\mathbf{A}_{n,k}(1,\cdot),\frac{1}{n}\|\xi^{(n)}\|_{2}^{2},\frac{1}{n}\sum_{i=1}^{n}\mathbf{r}(\xi_{i})\right),\quad
n\in\mathbb{N}.$

Since by part (iv) of Assumption 2.1, the domain of $\bar{\Lambda}$, the log
mgf of $(\xi_{1}^{2},\mathbf{r}(\xi_{1}))$, contains a neighborhood of the
origin, by Cramér’s theorem
$\\{\left(\|\xi^{(n)}\|_{2}^{2},\frac{1}{n}\sum_{i=1}^{n}\mathbf{r}(\xi_{i})\right)\\}_{n\in\mathbb{N}}$
satisfies an LDP in $\mathbb{R}^{2}$ with the convex GRF $\bar{\Lambda}^{*}$,
equal to the Legendre transform of $\bar{\Lambda}$.

The independence of $\mathbf{A}_{n,k}$ from $\\{\xi_{j}\\}_{j\in\mathbb{N}}$,
along with Theorem 3.4 of [5] (applied to the case of $p=2$ therein, with
their canonically projected $X^{(k)}$ equivalent to our
$\mathbf{A}_{n,k}(1,\cdot)$) then implies that the sequence
$\\{S^{(n)}\\}_{n\in\mathbb{N}}$ satisfies an LDP with the convex GRF
$J:\mathbb{R}^{k+2}\rightarrow[0,\infty]$ defined by

$J(a,b,c):=-\tfrac{1}{2}\log(1-\|a\|_{2}^{2})+\widehat{J}(b,c),$

for $a\in\mathbb{R}^{k}$ such that $\|a\|_{2}^{2}<1$ and $b,c\in\mathbb{R}$.
Then, by the contraction principle, $\\{R^{(n)}\\}_{n\in\mathbb{N}}$ satisfies
an LDP with the GRF $J_{R}:\mathbb{R}^{k+1}\rightarrow\mathbb{R}$ defined as
follows:

$J_{R}(x,z):=\inf_{y\in\mathbb{R}:y>\|x\|^{2}}J(xy^{-1/2},y,z),\quad
x\in\mathbb{R}^{k},z\geq 0.$

Note that $J_{R}$ is convex due to [10, Lemma 6.2] and Theorem 5.3 of [25].

Let $\Phi$ be as in (6.4) and $T$ be as in Assumption 2.1(ii). Fix
$t_{1}\in\mathbb{R}^{k}$, $t_{2}<T$, and let $t=(t_{1},t_{2})$. For
$0<\delta<\infty$,

$\mathbb{E}[e^{\delta n\langle t,R^{(n)}\rangle}]=\mathbb{E}\left[e^{\delta
n\Psi_{\mathsf{L}_{n,k}}(t_{1},t_{2})}\right].$ (6.9)

Since $t_{2}<T$, due to Lemma 6.1 and the observation (6.9), we see that
$\limsup_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{E}[e^{\delta n\langle
t,R^{(n)}\rangle}]$ is finite. By Varadhan’s lemma [8, Theorem 4.3.1] and the
finiteness condition (4.3.3) therein applied to any $\delta>1$, we find that

$\Phi(t_{1},t_{2})=\lim_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{E}[e^{n\langle
t,R^{(n)}\rangle}]=\sup_{\tau_{1}\in\mathbb{R}^{k},\tau_{2}\in\mathbb{R}}\\{\langle
t_{1},\tau_{1}\rangle+t_{2}\tau_{2}-J_{R}(\tau_{1},\tau_{2})\\}.$ (6.10)

where the first equality above holds by Lemma 6.2 on observing that $\Phi_{n}$
in (6.3) is the scaled log mgf of $R_{n}$.

Now fix $t_{1}\in\mathbb{R}^{k}$ and $t_{2}\geq T$. We claim that (6.10)
continues to hold, but now with both sides equal to infinity. The fact that
$\Phi(t_{1},t_{2})=\infty$ follows from the definition (6.4) of $\Phi$ and the
observation that $\Phi_{\nu}(t_{1},t_{2})=\infty$ when $t_{2}\geq T$ (by
Remark 2.2). To show that the term on the right-hand side of (6.10) is also
equal to infinity, for $s_{2}\in\mathbb{R}$, define
$\widetilde{\Lambda}(s_{2}):=\Lambda(0,s_{2})$. Note that
$\widetilde{\Lambda}$ is the log mgf of $\mathbf{r}(\xi_{1})$. Due to
Assumption 2.1(iv), the domain of $\widetilde{\Lambda}$ contains a non-empty
neighborhood around 0, hence by Cramér’s theorem the sequence
$\\{\frac{1}{n}\sum_{i=1}^{n}\mathbf{r}(\xi_{i})\\}_{n\in\mathbb{N}}$
satisfies an LDP in $\mathbb{R}$ with GRF $\widetilde{\Lambda}^{*}$. However,
due to the contraction principle and the continuity of the projection map, we
also know that
$\widetilde{\Lambda}^{*}(\tau_{2})=\inf_{\tau_{1}\in\mathbb{R}^{k}}J_{R}(\tau_{1},\tau_{2})$
for all $\tau_{2}\in\mathbb{R}$. Note that this infimum is attained at some
$\tau_{1}^{*}\in\mathbb{R}^{k}$ because, as a good rate function, $J_{R}$ is
lower semicontinuous with compact level sets. Therefore, on the right-hand
side of (6.10), if $t_{2}\geq T$, then

$\displaystyle\sup_{\tau_{1}\in\mathbb{R}^{k},\tau_{2}\in\mathbb{R}}\\{\langle
t_{1},\tau_{1}\rangle+t_{2}\tau_{2}-J_{R}(\tau_{1},\tau_{2})\\}$
$\displaystyle\geq\sup_{\tau_{2}\in\mathbb{R}}\\{\langle
t_{1},\tau_{1}^{*}\rangle+t_{2}\tau_{2}-J_{R}(\tau_{1}^{*},\tau_{2})\\}=\langle
t_{1},\tau_{1}^{*}\rangle+\widetilde{\Lambda}(t_{2})=\Lambda(0,t_{2})=\infty,$

where the first equality used the identity
$(\widetilde{\Lambda}^{*})^{*}=\widetilde{\Lambda}.$ Hence, (6.10) holds for
_all_ $t_{1}\in\mathbb{R}^{k}$ and $t_{2}\in\mathbb{R}$.

Note that (6.10) shows that $\Phi=J_{R}^{*}$. Due to the convexity of $J_{R}$
and Legendre duality (see, e.g., [8, Lemma 4.5.8]), we have that
$J_{R}=\Phi^{*}$. Hence, applying the contraction principle to the LDP for
$\\{R^{(n)}\\}_{n\in\mathbb{N}}$, we find that the annealed rate function
$\mathcal{J}^{\sf{an}}$ of Theorem 2.6 can be written as

$\displaystyle\mathcal{J}^{\sf{an}}(x)$
$\displaystyle=\inf_{\tau_{1}\in\mathbb{R}^{k},\tau_{2}\in\mathbb{R}:\tau_{1}\rho(\tau_{2})=x}\Phi^{*}(\tau_{1},\tau_{2})$
$\displaystyle=\inf_{\tau_{1}\in\mathbb{R}^{k},\tau_{2}\in\mathbb{R}:\tau_{1}\rho(\tau_{2})=x}\inf_{\nu\in\mathcal{P}(\mathbb{R}^{k})}\\{\Psi_{\nu}^{*}(\tau_{1},\tau_{2})+\mathbb{H}_{k}(\nu)\\}$
$\displaystyle=\inf_{\nu\in\mathcal{P}(\mathbb{R}^{k})}\inf_{\tau_{1}\in\mathbb{R}^{k},\tau_{2}\in\mathbb{R}:\tau_{1}\rho(\tau_{2})=x}\\{\Psi_{\nu}^{*}(\tau_{1},\tau_{2})+\mathbb{H}_{k}(\nu)\\}$
$\displaystyle=\inf_{\nu\in\mathcal{P}(\mathbb{R}^{k})}\\{\mathcal{J}_{\nu}^{\sf{qu}}(x)+\mathbb{H}_{k}(\nu)\\},$

where the second equality invokes (6.5) of Lemma 6.3, and the last equality
relies on (2.7). This completes the proof of Theorem 2.7. ∎

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Division of Applied Mathematics
Brown University
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