# Sharp Asymptotics for $q$-Norms of Random Vectors
in High-Dimensional $\ell_{p}^{n}$-Balls

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

Sharp large deviation results of Bahadur–Ranga Rao type are provided for the
$q$-norm of random vectors distributed on the $\ell_{p}^{n}$-ball
${\mathbb{B}}^{n}_{p}$ according to the cone probability measure or the
uniform distribution for $1\leq q<p<\infty$, thereby furthering previous large
deviation results by Kabluchko, Prochno and Thäle in the same setting. These
results are then applied to deduce sharp asymptotics for intersection volumes
of different $\ell_{p}^{n}$-balls in the spirit of Schechtman and
Schmuckenschläger, and for the length of the projection of an
$\ell_{p}^{n}$-ball onto a line with uniform random direction. The sharp large
deviation results are proven by providing convenient probabilistic
representations of the $q$-norms, employing local limit theorems to
approximate their densities, and then using geometric results for asymptotic
expansions of Laplace integrals to integrate these densities and derive
concrete probability estimates.

###### Key words and phrases:

Asymptotic geometric analysis, Bahadur Ranga Rao, high-dimensional convexity,
intersection volume, $\ell_{p}^{n}$-balls, $\ell_{p}^{n}$-spheres, large
deviation principles, precise large deviations, sharp asymptotics, sharp large
deviations, strong large deviations, volume of convex bodies.

###### 2010 Mathematics Subject Classification:

Primary: 52A23, 60F10 Secondary: 46B09, 60D05

##  1\. Introduction

The study of convex bodies in high dimensions, known today as asymptotic
geometric analysis, has arisen from the local theory of Banach spaces, which
aimed at analyzing infinite-dimensional normed spaces via their finite-
dimensional substructures, such as their unit balls. Despite having its origin
in the realm of functional analysis, the field has since established itself in
its own right, considering problems also beyond the study of centrally
symmetric convex bodies that occur naturally as the unit balls of Banach
spaces. In high dimensions convex bodies exhibit certain regularities, such as
concentration of measure phenomena (see, e.g., [22]), which make it highly
useful to approach them from a probabilistic perspective. As pointed out in
[6], it might seem counter-intuitive to analyze something exhibiting
regularities from a probabilistic perspective, as probability concerns itself
with studying the nature of irregularity, i.e. randomness, of given
quantities. But as with well-known limit theorems from probability such as the
law of large numbers and the central limit theorem, with large sample sizes
(and analogously – with high dimensionality) random objects exhibit
interesting patterns well characterized in the language of probability and
vice versa. Many results analogous to those from classic probability have been
found for high-dimensional convex sets, such as the central limit theorem
(see, e.g., Anttila, Ball and Perissinaki [5], Klartag [34, 35]). For further
background on high-dimensional convexity, see [6, 11, 22, 23].

The $\ell_{p}^{n}$-ball ${\mathbb{B}}^{n}_{p}$, $n\in\mathbb{N}$, has been a
prominent object of study, as it is the unit ball of the (finite-dimensional)
sequence space $\ell_{p}^{n}$, and has been the subject of a multitude of
results. We will name only a select few and refer to the survey by Prochno,
Thäle and Turchi [40] for a comprehensive summary of classic and contemporary
results. Let us denote by $\mathbf{U}_{n,p}$ the uniform distribution on the
Euclidean $\ell_{p}^{n}$-ball ${\mathbb{B}}_{p}^{n}$ and by $\mathbf{C}_{n,p}$
the cone probability measure on the $\ell_{p}^{n}$-sphere
${\mathbb{S}}_{p}^{n-1}$. Schechtman and Zinn [45] and Rachev and Rüschendorf
[41] showed a generalization of the Poincaré–Maxwell–Borel lemma, proving
that, for $k\in\mathbb{N}$ with $k<n$, the $k$-dimensional marginal
distribution of a random vector distributed according to $\mathbf{C}_{n,p}$
converges to a $k$-dimensional generalized Gaussian distribution as $n$
increases. They also provided a probabilistic representation for such random
vectors in terms of these generalized Gaussian distributions, which will be a
key building block in our main results. The primary quantity of interest of
this paper however is the behaviour of the $q$-norm $\|Z\|_{q}$ of a random
vector $Z$ in ${\mathbb{S}}_{p}^{n-1}$ and ${\mathbb{B}}_{p}^{n}$. This was
first studied by Schechtman and Zinn [45], who derived concentration
inequalities for $\|Z\|_{q}$ with $Z\sim\mathbf{C}_{n,p}$ and
$Z\sim\mathbf{U}_{n,p}$ for $q>p$. This is closely related to the intersection
volume of $t$-multiples of volume-normalized $\ell_{p}^{n}$-balls
${\mathbb{D}}_{p}^{n}:=\mathrm{vol}_{n}({\mathbb{B}}_{p}^{n})^{-1/n}{\mathbb{B}}_{p}^{n}$,
i.e. $\text{vol}_{n}({\mathbb{D}}_{p}^{n}\cap t{\mathbb{D}}_{q}^{n})$ with
$t\in[0,\infty)$, for which Schechtman and Schmuckenschläger [44] gave the
asymptotics for $t\neq 1$. Schechtman and Zinn [46] expanded their previous
results in [45], by not only considering the $q$-norm, but also images of
random vectors under Lipschitz functions in general. Thus, they gave
concentration inequalities for $f(Z)$, with $Z\sim\mathbf{C}_{n,p}$ and
$Z\sim\mathbf{U}_{n,p}$, $p\in[1,2)$, and $f$ a Lipschitz function with
respect to the Euclidean norm. Schmuckenschläger [47] provided a central limit
theorem (CLT) for $\|Z\|_{q}$ with $Z\sim\mathbf{C}_{n,p}$ and
$Z\sim\mathbf{U}_{n,p}$ and used it to refine the previous intersection
results in [44] for all $t\in(0,\infty)$. Naor [38] gave concentration
inequalities for $\|Z\|^{q}_{q}$ with $Z\sim\mathbf{C}_{n,p}$, showed that the
total variation distance between $\mathbf{C}_{n,p}$ and the normalized surface
measure $\sigma_{n,p}$ on ${\mathbb{S}}_{p}^{n-1}$ tends to zero proportional
to $n^{-1/2}$, and used the previously mentioned results to show a
concentration inequality for $\|Z\|^{q}_{q}$ with $Z\sim\sigma_{n,p}$. He also
discussed how concentration results similar to Schechtman and Zinn [46] for
$\|Z\|_{q}$ could already be derived from previous results of Gromov and
Milman [21] for the concentration of Lipschitz functions on convex bodies.
Kabluchko, Prochno and Thäle [28] gave a multivariate CLT for
$(\|Z\|_{q_{1}},\ldots,\|Z\|_{q_{d}})$ with $Z\sim\mathbf{U}_{n,p}$ in the
spirit of [47] and also considered the asymptotics for the intersection volume
of multiple $\ell_{p}^{n}$-balls, i.e.
$\text{vol}_{n}({\mathbb{D}}_{p}^{n}\cap
t_{1}{\mathbb{D}}_{q_{1}}^{n}\cap\cdots\cap t_{d}{\mathbb{D}}_{q_{d}}^{n})$
with $t_{i}\in[0,\infty)$. This CLT was furthermore applied by the same
authors to infer a central limit theorem for the length of
${\mathbb{B}}_{p}^{n}$ projected onto a line with uniform random direction.
Moreover, they provided a large deviation principle (LDP) for $\|Z\|_{q}$ with
$Z\sim\mathbf{C}_{n,p}$ and $Z\sim\mathbf{U}_{n,p}$. In a follow-up paper
[30], the same authors showed a CLT for $\|Z\|_{q}$, where the distribution of
$Z$ is taken from a wider class of $p$-radial distributions
$\mathbf{P}_{n,p,\textbf{W}}$, introduced by Barthe, Guédon, Mendelson and
Naor [8], consisting of mixtures of $\mathbf{U}_{n,p}$ and $\mathbf{C}_{n,p}$,
combined via a measure W on $[0,\infty)$. This class contains both
$\mathbf{U}_{n,p}$ and $\mathbf{C}_{n,p}$, but also distributions
corresponding with geometrically interesting projections (see, e.g., [30,
Introduction, (iii)]). Finally, they gave a moderate and a large deviation
principle for $\|Z\|_{q}$ with $Z\sim\mathbf{P}_{n,p,\textbf{W}}$.

Generally, studying large deviations within asymptotic geometric analysis has
started fairly recently with Gantert, Kim and Ramanan [19], who gave an LDP
for projections of random points in $\ell_{p}^{n}$-balls with distributions
$\mathbf{C}_{n,p}$ and $\mathbf{U}_{n,p}$ onto both random and fixed one-
dimensional subspaces. Today, large deviations theory has become a well-
established toolbox in high-dimensional convex geometry, giving rise to a
plethora of results (see, e.g., [3, 4, 28, 29, 30, 32, 33]). Recently, a new
tool from large deviations theory was introduced to asymptotic geometric
analysis by Liao and Ramanan [37]. They gave sharp large deviation (SLD)
results in the spirit of Bahadur and Ranga Rao [7] and Petrov [39] for the
projections of random points in $\ell_{p}^{n}$-balls with distributions
$\mathbf{C}_{n,p}$ and $\mathbf{U}_{n,p}$ onto a fixed one-dimensional
subspace. Other works in asymptotic geometric analysis have also employed
methods from sharp large deviations theory as well, such as Kabluchko and
Prochno [27], who derived asymptotic volumes for generalizations of
$\ell_{p}^{n}$-balls, known as Orlicz balls, and showed a Schechtman and
Schmuckenschläger-type result by considering intersection volumes of Orlicz
balls. Their results on Orlicz balls were then expanded upon by Alonso-
Guiterréz and Prochno in [2], who gave the exact asymptotic volume of Orlicz
balls and provided thin-shell concentrations for them, augmenting their
results into sharp asymptotics under certain conditions.While LDPs only give
tail asymptotics on a logarithmic scale, the sharp asymptotics provided by
sharp large deviations theory can give tail estimates for concrete values of
$n\in\mathbb{N}$, which makes them significantly more useful for practical
applications. Moreover, a lot of idiosyncrasies of the underlying
distributions, that are drowned out on the LDP scale, are still visible on the
SLD scale, thus giving a deeper understanding of the geometric interpretation
of the quantities involved. This paper will follow closely in the footsteps of
Liao and Ramanan [37] and establish SLD results for the $q$-norms of random
vectors with distribution $\mathbf{C}_{n,p}$ and $\mathbf{U}_{n,p}$.
Furthermore, we will use these results to expand on works of Schechtman and
Schmuckenschläger [44], Schmuckenschläger [47], and Kabluchko, Prochno and
Thäle [28] for intersection volumes of $\ell_{p}^{n}$-balls by giving sharp
asymptotics for $\text{vol}_{n}({\mathbb{D}}_{p}^{n}\cap
t{\mathbb{D}}_{q}^{n})$ at a considerably improved rate for $1\leq q<p<\infty$
and $t>C(p,q)$ bigger than some constant dependent on $p$ and $q$ only.
Additionally, we will also apply our results for $\ell_{p}^{n}$-spheres to
retain sharp asymptotics for the length of the projection of an
$\ell_{p}^{n}$-ball onto the line spanned by a uniform random direction.

The paper will proceed as follows: in Section 2 some basic notation and
definitions will be provided while also giving some appropriate background on
the involved large deviations theory. Furthermore, we will recapitulate some
existing results that are relevant to this paper. In Section 3 we will present
our main results regarding the $q$-norms of random vectors on
$\ell_{p}^{n}$-spheres and $\ell_{p}^{n}$-balls. Also, we will present and
prove their application to intersections and one-dimensional projections of
$\ell_{p}^{n}$-balls, and outline the idea of the two central proofs. In
Section 4 we will reformulate the target probabilities from the main results
in terms of useful probabilistic representations, using well-established
representations of random vectors in $\ell_{p}^{n}$-balls of Schechtman and
Zinn [45] and Rachev and Rüschendorf [41]. In Section 5 local density
approximations of these probabilistic representations will be provided. In
Sections 6 and 7 we will prove the SLD results for $\ell_{p}^{n}$-spheres and
$\ell_{p}^{n}$-balls, respectively, by integrating over the density estimates.
For that, we will utilize some geometric results for asymptotic expansions of
Laplace integrals from Adriani and Baldi [1] and Breitung and Hohenbichler
[12].

##  2\. Preliminaries

###  2.1. Notation and important distributions

We denote by $\mathrm{vol}_{d}$ the $d$-dimensional Lebesgue measure on
$\mathbb{R}^{d}$ and write $\mathcal{B}(\mathbb{R}^{d})$ for the
$\sigma$-field of Borel sets in $\mathbb{R}^{d}$. For a set
$A\in\mathcal{B}(\mathbb{R}^{d})$ we write $A^{\circ},\overline{A},$ $\partial
A$, and $A^{c}$ for the interior, closure, boundary and complement of $A$,
respectively. Furthermore, we write $\langle\,\cdot\,,\,\cdot\,\rangle$ for
the standard scalar product in $\mathbb{R}^{d}$. For
$g:\mathbb{R}^{d}\to\mathbb{R}^{d}$, we denote by $J_{x}g(x^{*})$ the Jacobian
of $g$ with respect to the vector $x$ evaluated at $x^{*}\in\mathbb{R}^{d}$,
and for $f:\mathbb{R}^{d}\to\mathbb{R}$ by $\nabla_{x}f(x^{*})$ and
$\mathcal{H}_{x}f(x^{*})$ the gradient and Hessian of $f$ with respect to the
vector $x$ evaluated at $x^{*}\in\mathbb{R}^{d}$, respectively, and use the
shorthand notation

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

We write $(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}$ for a standard column vector
and for $x,y\in\mathbb{R}^{d}$, we write their product $x^{T}y$ as $xy$,
skipping the explicit transpose for brevity of notation. Given a random
variable $X$ with distribution $\mathbf{P}$, we write $X\sim\mathbf{P}$ and
denote by $\mathbb{E}X$ its expectation. For two random variables $X,Y$ with
the same distribution we write $X\overset{d}{=}Y$. For a random vector $X$ in
$\mathbb{R}^{d}$ and $s\in\mathbb{R}^{d}$, denote by
$\varphi_{X}(s):=\mathbb{E}[e^{\langle s,X\rangle}]$ and
$\Lambda_{X}(s):=\log\varphi_{X}(s)$ the moment generating function and
cumulant generating function (m.g.f. and c.g.f.), respectively. We call the
set of $s\in\mathbb{R}^{d}$ for which $\Lambda_{X}(s)<\infty$ the effective
domain $\mathcal{D}_{X}$ of $\Lambda_{X}$. Moreover, for $x\in\mathbb{R}^{d}$
we denote by $\Lambda_{X}^{*}(x):=\sup_{s\in\mathbb{R}^{d}}[\langle
x,s\rangle-\Lambda_{X}(s)]$ the Legendre-Fenchel transform of the c.g.f.
$\Lambda_{X}$. When considering sequences in $n\in\mathbb{N}$, we denote by
$o(1)$ a sequence that tends to zero as $n\to\infty$.

Let us consider the class of distributions at the core of the probabilistic
constructions throughout this paper. We say a real-valued random variable $X$
has a generalized Gaussian distribution if its distribution has Lebesgue
density

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

where $m\in\mathbb{R}$ and $a,b>0$, and denote this by
$X\sim{\mathbf{N}}_{\textup{gen}}(m,a,b)$. As mentioned in the introduction,
the generalized Gaussian distributions are essential for constructing
probabilistically equivalent representations of the quantities of interest,
based on results of Schechtman and Zinn [45] and Rachev and Rüschendorf [41].
For these constructions we will be using the specific generalized Gaussian
distribution
$\mathbf{N}_{p}:={\mathbf{N}}_{\textup{gen}}\left(0,p^{1/p},p\right)$ ,
$p\in[1,\infty),$ with density

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

For $X\sim\mathbf{N}_{p}$ and $r>0$, we write $M_{p}(r):=\mathbb{E}|X|^{r}$
for the $r$-th absolute moment of $X$, for which it holds that

$M_{p}(r):=\mathbb{E}|X|^{r}=\left(\frac{p^{r/p}}{r+1}\,\frac{\Gamma\left(1+\frac{r+1}{p}\right)}{\Gamma\left(1+\frac{1}{p}\right)}\right).$
(2)

###  2.2. Background material from (sharp) large deviations theory

We will give some basic notions and definitions from large deviations theory.
To keep this paper self-contained, we will present them here, while referring
the reader to [16, 17, 31] for additional background material on large
deviations. Furthermore, we want to give some insight into the methods of the
lesser known theory of sharp large deviations.

######  Definition 2.1.

Let $(\textup{{P}}_{n})_{n\in\mathbb{N}}$ be a sequence of probability
measures on $\mathbb{R}^{d}$. We say that
$(\textup{{P}}_{n})_{n\in\mathbb{N}}$ satisfies a large deviation principle
(LDP) if there are two functions $s:\mathbb{N}\to\mathbb{R}$ and
$\mathcal{I}:\mathbb{R}^{d}\to[0,\infty]$, such that $\mathcal{I}$ is lower
semi-continuous and

$\begin{array}[]{lllll}a)&\displaystyle\underset{n\to\infty}{\textup{lim
sup}}\,\,\frac{1}{s(n)}\log\textup{{P}}_{n}(C)&\leq&-\mathcal{I}(C)&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{
for all }\,\,C\subset\mathbb{R}^{n}\text{ closed,}\\\
b)&\displaystyle\underset{n\to\infty}{\textup{lim
inf}}\,\,\frac{1}{s(n)}\log\textup{{P}}_{n}(O)&\geq&-\mathcal{I}(O)&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{
for all }\,\,O\subset\mathbb{R}^{n}\text{ open,}\\\ \end{array}$

where for $B\subset\mathbb{R}^{d}$ we define $\mathcal{I}(B):=\inf_{x\in
B}\mathcal{I}(x)$. We call $s$ the speed and $\mathcal{I}$ the rate function.
We say that $\mathcal{I}$ is a good rate function, if it has compact sub-level
sets.

We apply the definition of LDPs to sequences of random variables as well by
applying the above definition to the sequence of their distributions. In our
setting the sequence parameter $n\in\mathbb{N}$ will furthermore coincide with
the space dimension $d\in\mathbb{N}$, as we are considering the effects of
increasing dimensionality. Given a sequence $(X^{(n)})_{n\in\mathbb{N}}$ of
i.i.d. random vectors in $\mathbb{R}^{d}$, one is frequently interested in the
behaviour of the sequence $(S^{(n)})_{n\in\mathbb{N}}$ of the empirical
averages $S^{(n)}:=\frac{1}{n}\sum_{i=1}^{n}X^{(i)}\in\mathbb{R}^{d}$. One of
the most well known and most frequently used results in the theory of large
deviations is the theorem of Cramér, which states that if the c.g.f.
$\Lambda_{X}$ is finite in an open neighbourhood of the origin, then
$(S^{(n)})_{n\in\mathbb{N}}$ satisfies an LDP in $\mathbb{R}^{d}$ with speed
$n$ and rate function $\Lambda^{*}_{X}$ (see, e.g., [16, Theorem 2.2.30,
Theorem 6.1.3, Corollary 6.1.6]). Hence, under suitable exponential moment
assumptions for the $X^{(n)}$, we can already infer the large deviation
behaviour of $(S^{(n)})_{n\in\mathbb{N}}$.

The classic LDP gives us an idea of the asymptotic deviation behaviour of a
sequence of distributions on a logarithmic scale. By doing this however, a lot
of subtleties of the underlying distributions can be drowned out. Many small
and medium scale properties of a given sequence of distributions are often
missed in the asymptotic analysis of LDPs, since they either disappear for
very large $n\in\mathbb{N}$ or are drowned out by other, more significant
phenomena of the distribution. Thus, one is also interested in considering
large deviations on a nonlogarithmic scale, which we refer to as “sharp” large
deviations (also called “precise” or “strong” large deviations in the
literature). One of the first and most prominent results in this regard was
shown by Bahadur and Ranga Rao [7]. They showed that for a sequence
$(X^{(n)})_{n\in\mathbb{N}}$ of i.i.d. random variables and any
$z>\mathbb{E}[X^{(n)}]$ with $\Lambda_{X}^{*}(z)<\infty$, it holds that

$\mathbb{P}\left(S^{(n)}>z\right)=\frac{1}{\sqrt{2\pi
n}\,\kappa(z)\xi(z)}e^{-n\Lambda_{X}^{*}(z)}(1+\textit{o}(1)),$

where $\kappa(z)$ and $\xi(z)$ are only dependent on the distribution of the
$X^{(n)}$ and the deviation size $z$. This is proven via a (somewhat implicit)
application of the the so-called saddle point method (or method of steepest
descents), which was established by Debye [15], and brought to the realm of
probability by Esscher [18] and Daniels [14]. The saddle point method
generalizes Laplace’s method for integral approximation to the complex plane,
and is therefore highly useful when dealing with integrals over characteristic
functions. In general, for appropriate functions $f,g$ and $n\in\mathbb{N}$
large, the saddle point method gives a way to approximate Laplace-type
integrals $\int_{P}g(z)e^{-nf(z)}\textup{d}z$ along complex paths $P$, by
deforming the integration path using Cauchy’s theorem, into some $\tilde{P}$
that passes through a saddle point of $f$. The mass of the reformulated
integral is then heavily concentrated around the saddle point and standard
integral expansion methods, such as Edgeworth expansion, can be used to great
effect. In the realm of probability, this has been used for both tail
probabilities (e.g. Esscher [18], Cramér [13]) and densities of random
variables (e.g. Daniels [14], Richter [42, 43]), by writing them as an
integral over their characteristic functions, using the Fourier inversion
formula, and then approximating those integrals via the use of a complex
saddle point. We say that this was used “somewhat implicitly” in certain
results, such as those of Esscher [18], Cramér [13] and Bahadur and Ranga Rao
[7], since the technique used therein, which is a certain change of measure,
often called exponential tilting or Esscher/Cramér transform, under the
surface employs saddle points as well. For further background on this method,
we refer to the book of Jensen [26].

As mentioned in the introduction, Section 5 will provide density estimates for
our probabilistic representations from Section 4, which are derived using the
saddle point method. However, since our probabilistic representations are
given as sums of i.i.d. random vectors, we will refer to previous results
where this was done explicitly, while making sure that the conditions for
their application are still met in our setting. Generally, the core idea of
the saddle point method, which is reformula-ting an integral such that all of
its mass heavily concentrates around a critical point, around which we can
then employ approximation methods, is used in the overall proof of our main
results in a broader sense as well. We reformulate our target probabilities
via some convenient representations, whose densities we also provide, such
that the remaining integrals then heavily concentrate their mass around a
given critical point, such that approximations at that point yield accurate
results, as we will see in Sections 6 and 7.

###  2.3. Distributions on $\ell_{p}^{n}$-balls

For $p\in[1,\infty]$, $n\in\mathbb{N}$, and
$x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$ we denote by

$\|x\|_{p}:=\begin{cases}\Big{(}\sum\limits_{i=1}^{n}|x_{i}|^{p}\Big{)}^{1/p}&:p<\infty\\\
\max\\{|x_{1}|,\ldots,|x_{n}|\\}&:p=\infty\end{cases}$ (3)

the $\ell_{p}^{n}$-norm of $x$. Let
${\mathbb{B}}_{p}^{n}:=\\{x\in\mathbb{R}^{n}:\|x\|_{p}\leq 1\\}$ be the unit
$\ell_{p}^{n}$-ball and
${\mathbb{S}}_{p}^{n-1}:=\\{x\in\mathbb{R}^{n}:\|x\|_{p}=1\\}$ be the unit
$\ell_{p}^{n}$-sphere. We define the uniform distribution on
${\mathbb{B}}^{n}_{p}$ and cone probability measure on
${\mathbb{S}}^{n-1}_{p}$ as

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

The following result is the basis of our probabilistic representations for
random vectors with distribution $\mathbf{C}_{n,p}$ and $\mathbf{U}_{n,p}$ and
is due to [41] and [45].

######  Lemma 2.2.

Let $p\in[1,\infty)$, $Y=(Y_{1},\ldots,Y_{n})$ be a random vector in
$\mathbb{R}^{n}$ with $Y_{i}\sim\mathbf{N}_{p}$ i.i.d. , and $U$ an
independent random variable uniformly distributed on $[0,1]$. Then,

  * i)

the random vector ${Y}/{\|Y\|_{p}}$ has distribution $\mathbf{C}_{n,p}$ and is
independent of $\|Y\|_{p},$

  * ii)

the random vector $U^{1/n}\,{Y}/{\|Y\|_{p}}$ has distribution
$\mathbf{U}_{n,p}$.

###  2.4. LDPs for $q$-norms in $\ell_{p}^{n}$-balls

Throughout this paper we assume $1\leq q<p<\infty$. The main variables of
interest will be the $q$-norms of the random vectors
$Z^{(n)},\mathscr{Z}^{(n)}\in{\mathbb{B}}^{n}_{p}$ with
$Z^{(n)}\sim\mathbf{C}_{n,p}$ and $\mathscr{Z}^{(n)}\sim\mathbf{U}_{n,p}$.
Note, that we will always denote quantities related to
$\mathscr{Z}^{(n)}\sim\mathbf{U}_{n,p}$ cursively. To get non-trivial results,
our target variables also need to be appropriately rescaled. Thus, for random
vectors $Z^{(n)},\mathscr{Z}^{(n)}\in{\mathbb{B}}^{n}_{p}$ with
$Z^{(n)}\sim\mathbf{C}_{n,p}$ and $\mathscr{Z}^{(n)}\sim\mathbf{U}_{n,p}$, our
target variables will be $n^{1/p-1/q}\,\|Z^{(n)}\|_{q}$ and
$n^{1/p-1/q}\,\|\mathscr{Z}^{(n)}\|_{q}$, respectively. We set

$\|Z\|:=\left(n^{1/p-1/q}\,\|Z^{(n)}\|_{q}\right)_{n\in\mathbb{N}}\qquad\text{
and
}\qquad\|\mathscr{Z}\|:=\left(n^{1/p-1/q}\,\|\mathscr{Z}^{(n)}\|_{q}\right)_{n\in\mathbb{N}}.$

It follows via the strong law of large numbers and the continuous mapping
theorem applied to the probabilistic representations in (4) and (5) that the
expectations of $\|Z\|$ and $\|\mathscr{Z}\|$ converge in $n\in\mathbb{N}$ to
$m_{p,q}:=M_{p}(q)^{1/q}$. For fixed $n\in\mathbb{N}$ we will denote

$\mathbb{E}\left[n^{1/p-1/q}\,\|Z^{(n)}\|_{q}\right]:=m_{n,p,q}\qquad\text{
and
}\qquad\mathbb{E}\left[n^{1/p-1/q}\,\|\mathscr{Z}^{(n)}\|_{q}\right]:=\mathscr{m}_{n,p,q}.$

Furthermore, LDPs for $\|Z\|$ and $\|\mathscr{Z}\|$ have been given in
previous works, which we want to include here explicitly. But first, let us
look at the following probabilistic representations of $\|Z\|$ and
$\|\mathscr{Z}\|$, since the LDPs are given with respect to the c.g.f. of
these representations: Let $(Y^{(n)})_{n\in\mathbb{N}}$ be a sequence of
i.i.d. random vectors $Y^{(n)}:=(Y^{(n)}_{1},\ldots,Y^{(n)}_{n})$ with
$Y^{(n)}_{i}\sim\mathbf{N}_{p}$, and $U$ a random variable, independent of the
$Y^{(n)}_{i}$, and uniformly distributed on $[0,1]$. Then, we can see via
Lemma 2.2 that

$n^{1/p-1/q}\,\|Z^{(n)}\|_{q}\overset{d}{=}n^{1/p-1/q}\,\frac{\|Y^{(n)}\|_{q}}{\|Y^{(n)}\|_{p}}=\frac{{\left(\frac{1}{n}\sum_{i=1}^{n}{|Y^{(n)}_{i}|}^{q}\right)}^{1/q}}{{\left(\frac{1}{n}\sum_{i=1}^{n}{|Y^{(n)}_{i}|}^{p}\right)}^{1/p}},$
(4)

and

$n^{1/p-1/q}\,\|\mathscr{Z}^{(n)}\|_{q}\overset{d}{=}n^{1/p-1/q}\,U^{1/n}\,\frac{\|Y^{(n)}\|_{q}}{\|Y^{(n)}\|_{p}}=U^{1/n}\,\frac{{\left(\frac{1}{n}\sum_{i=1}^{n}{|Y^{(n)}_{i}|}^{q}\right)}^{1/q}}{{\left(\frac{1}{n}\sum_{i=1}^{n}{|Y^{(n)}_{i}|}^{p}\right)}^{1/p}}.$
(5)

Define

$V^{(n)}:=\left(V^{(n)}_{1},\ldots,V^{(n)}_{n}\right)\in\mathbb{R}^{2n}\hskip
2.84544pt\qquad\text{ with }\qquad
V^{(n)}_{i}:=\left({|Y_{i}^{(n)}|}^{q},{|Y_{i}^{(n)}|}^{p}\right),\,\,\quad$
(6)

and

$\quad\mathscr{V}^{(n)}:=\left(\mathscr{V}^{(n)}_{1},\ldots,\mathscr{V}^{(n)}_{n}\right)\in\mathbb{R}^{3n}\qquad\text{
with
}\qquad\mathscr{V}^{(n)}_{i}:=\left({|Y_{i}^{(n)}|}^{q},{|Y_{i}^{(n)}|}^{p},U^{1/n}\right).$
(7)

We denote the m.g.f. and c.g.f. of the $V^{(n)}_{i}$ as

$\displaystyle\varphi_{p}(\tau):=\int_{\mathbb{R}}e^{tau_{1}|y|^{q}+\tau_{2}|y|^{p}}f_{p}(y)\,\textup{d}y\qquad\text{
and
}\qquad\Lambda_{p}(\tau):=\log\int_{\mathbb{R}}e^{\tau_{1}|y|^{q}+\tau_{2}|y|^{p}}f_{p}(y)\,\textup{d}y,$
(8)

for $\tau=(\tau_{1},\tau_{2})\in\mathbb{R}^{2}$ and the Legendre-Fenchel
transform of $\Lambda_{p}$ as

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

Let $\mathcal{D}_{p}$ be the effective domain of $\Lambda_{p}$. Since $q<p$,
for the integral in both $\varphi_{p}$ and $\Lambda_{p}$ to be finite, the
sign of the dominant term in the exponent must be negative. Remembering the
definition of $f_{p}$, one can see that this is given for $tau_{2}<{1}/{p}$,
thus $\mathcal{D}_{p}=\mathbb{R}\times\left(-\infty,1/p\right)$. Now, we want
to characterize the points $x\in\mathbb{R}^{2}$ for which there exists a
$\tau(x)\in\mathcal{D}_{p}$, such that

$\Lambda_{p}^{*}(x)=\langle x,\tau(x)\rangle-\Lambda_{p}(\tau(x)),$ (9)

i.e. for which the function $g_{x}(\tau):=\langle
x,\tau\rangle-\Lambda_{p}(\tau)$ does not attain its supremum at infinity. We
will do this along the lines of [1, Section 2]. It holds that $\Lambda_{p}$ is
convex in $\tau$ (see standard properties of the c.g.f. in e.g. [16, Lemma
2.2.31]), hence $g_{x}(\tau):=\langle x,\tau\rangle-\Lambda_{p}(\tau)$ is
concave as a sum of concave functions. Then, for a given $x\in\mathbb{R}^{2}$,
there either exists a $\tau(x)\in\mathbb{R}^{2}$ that satisfies
$x=\nabla_{\tau}\,\Lambda_{p}(\tau)$, i.e. that is a root of
$\nabla_{\tau}\,g_{x}(\tau)$, or the supremum of $g_{x}$ will be attained at
infinity. If such a $\tau(x)$ exists and lies in $\mathcal{D}_{p}$, it holds
that $\Lambda_{p}^{*}(x)=\langle x,\tau(x)\rangle-\Lambda_{p}(\tau(x))<\infty$
(see [16, Lemma 2.2.31]). Since the $V_{i}^{(n)}$ are not concentrated on a
hyperplane (as $1\leq q<p<\infty$), the covariance matrix of their
distribution given by $\mathcal{H}_{\tau}\,\Lambda_{p}(0,0)$ is positive
definite and thereby invertible. $\mathcal{H}_{\tau}\,\Lambda_{p}(\tau)$ for
$\tau\in\mathcal{D}_{p}$ can also be interpreted as the covariance matrix of
an exponentially shifted distribution of $V_{i}^{(n)}$ (see [1, p. 374]),
which, by the same argument, is also not concentrated on any hyperplane, hence
the $\mathcal{H}_{\tau}\,\Lambda_{p}(\tau)$ are positive definite as well.
This implies that $\Lambda_{p}(\tau)$ is strictly convex on $\mathcal{D}_{p}$,
thereby also making $g_{x}$ strictly concave on $\mathcal{D}_{p}$. Hence, the
strict concavity of $g_{x}$ then ensures that $\tau(x)$ is unique in
$\mathcal{D}_{p}$ in the above described property. We denote the set of
$x\in\mathbb{R}^{2}$ for which such a $\tau(x)\in\mathcal{D}_{p}$ exists as
$\mathcal{J}_{p}$ and call it the admissible domain of $\Lambda_{p}^{*}$.

######  Remark 2.3.

Note that the admissible domain $\mathcal{J}_{p}$ is the image of
$\mathcal{D}_{p}$ under the derivative of the c.g.f. $\Lambda_{p}$. It
actually holds that $\nabla_{\tau}\Lambda_{p}(\tau)$ is a bijection from the
interior of the effective domain of $\Lambda_{p}$ into the interior of the
effective domain of $\Lambda_{p}^{*}$, by the properties of the Legendre
transform (see [Rockafellar, Theorem 26.5]). Since $\mathcal{D}_{p}$ is open
and $\nabla_{\tau}\Lambda_{p}$ continuous on $\mathcal{D}_{p}$, we thereby get
that the effective domain of $\Lambda_{p}^{*}$ is also open and thus,
$\mathcal{J}_{p}$ is simply the effective domain of $\Lambda_{p}^{*}$.

For the sequence $\|Z\|$ the following LDP has already been shown by
Kabluchko, Prochno and Thäle [28, Section 5.1]:

######  Proposition 2.4.

Let $1\leq q<p<\infty$ and $Z^{(n)}\sim\mathbf{C}_{n,p}$ be a random vector in
${\mathbb{S}}^{n-1}_{p}$. Then the sequence
$\left(n^{1/p-1/q}\,\|Z^{(n)}\|_{q}\right)_{n\in\mathbb{N}}$ satisfies an LDP
with speed $n$ and good rate function

$\mathcal{I}_{\|Z\|}(z):=\begin{cases}\displaystyle\inf_{{\text{$t_{1},t_{2}>0$}}\atop{\text{$t_{1}^{1/q}t_{2}^{-1/p}=z$}}}\Lambda_{p}^{*}(t_{1},t_{2})&:z>0\\\
+\infty&:z\leq 0.\end{cases}$

In [37, Lemma 2.1, Appendix A] Liao and Ramanan established a simplification
of a similar rate function in a different setting. Their arguments can be
analogously applied in our setting to derive the following result:

######  Lemma 2.5.

Let $z>m_{p,q}$ such that $z^{*}:=(z^{q},1)\in\mathcal{J}_{p}$. Then

$\displaystyle\mathcal{I}_{\|Z\|}(z)=\inf_{{\text{$t_{1},t_{2}>0$}}\atop{\text{$t_{1}^{1/q}t_{2}^{-1/p}=z$}}}\Lambda_{p}^{*}(t_{1},t_{2})=\Lambda_{p}^{*}(z^{*}),$

with $z^{*}$ being the unique point at which $\Lambda_{p}^{*}$ attains its
infimum under the above conditions.

To keep this paper self-contained, we will present the analogous proof of this
in the Appendix. For the sequence $\|\mathscr{Z}\|$, the following LDP was
also provided by Kabluchko, Prochno and Thäle in [28, Theorem 1.2]:

######  Proposition 2.6.

Let $1\leq q<p<\infty$ and $\mathscr{Z}^{(n)}\sim\mathbf{U}_{n,p}$ be a random
vector in ${\mathbb{B}}^{n}_{p}$. Then the sequence
$\left(n^{1/p-1/q}\,\|\mathscr{Z}^{(n)}\|_{q}\right)_{n\in\mathbb{N}}$
satisfies an LDP with speed $n$ and good rate function

$\mathcal{I}_{\|\mathscr{Z}\|}(z):=\begin{cases}\displaystyle\inf_{{\text{$z=z_{1}z_{2}$}}\atop{\text{$z_{1},z_{2}>0$}}}\big{[}\mathcal{I}_{\|Z\|}(z_{1})+\mathcal{I}_{U}(z_{2})\big{]}&:z>0\\\
+\infty&:z\leq 0,\end{cases}$

with $\mathcal{I}_{\|Z\|}$ as in Proposition 2.4 and

$\mathcal{I}_{U}(z_{2}):=\begin{cases}\displaystyle-\log(z_{2})&:z_{2}\in(0,1]\\\
+\infty&:otherwise.\end{cases}$

We again show that the above infimum is attained at a unique point satisfying
the infimum condition.

######  Lemma 2.7.

Assume the same setting as in Proposition 2.6. For $z>m_{p,q}$, we can
simplify the rate function by combining the two infimum operations to get

$\mathcal{I}_{\|\mathscr{Z}\|}(z)=\displaystyle\inf_{{\text{$z=t_{1}^{1/q}t_{2}^{-1/p}t_{3}$}}\atop{\text{$t_{1},t_{2}>0,t_{3}\in(0,1]$}}}\big{[}\Lambda_{p}^{*}(t_{1},t_{2})-\log(t_{3})\big{]}.$

We define

$\mathcal{I}_{\mathscr{S}}(t):=\big{[}\Lambda_{p}^{*}(t_{1},t_{2})-\log(t_{3})\big{]},\qquad
t_{1},t_{2}\in\mathbb{R},\quad t_{3}\in(0,1],$

and set $z^{*}:=(z^{q},1)\in\mathbb{R}^{2}$,
$z^{**}:=(z^{q},1,1)\in\mathbb{R}^{3}$. It then holds for $z>m_{p,q}$ with
$z^{*}\in\mathcal{J}_{p}$ that

$\displaystyle\mathcal{I}_{\|\mathscr{Z}\|}(z)=\mathcal{I}_{\mathscr{S}}(z^{**})=\Lambda_{p}^{*}(z^{*}),$

with $z^{**}$ being the unique point at which $\mathcal{I}_{\mathscr{S}}$
attains its infimum under the above conditions.

Thus, for $z>m_{p,q}$ with $z^{*}\in\mathcal{J}_{p}$ both $\|Z\|$ and
$\|\mathscr{Z}\|$ satisfy LDPs with the same speed and rate function. Again,
the proof of this is relegated to the Appendix.

######  Remark 2.8.

Note that in the results within this paper, deviations from the ”limit
expectation” $m_{p,q}$ are considered, even though the sequences $\|Z\|$ and
$\|\mathscr{Z}\|$ have respective expectations $m_{n,p,q}$ and
$\mathscr{m}_{n,p,q}$, that only converge to $m_{p,q}$ in $n\in\mathbb{N}$.
This, however, is not an issue for our results. As seen in (4) and (5), the
sequences are represented via the empirical averages of probabilistic
representations seen in (6) and (7). The expectations of these representations
only ever play a role in our proofs regarding the behaviour of the
corresponding c.g.f.s, specifically only in the case of $\|Z\|$ (e.g. in the
proofs of Lemma 2.7 and Lemma 4.1 or implicitly in the proof of the density
approximations in Section 5). As the $V_{i}^{(n)}$ in (6) are i.i.d., they all
share the same c.g.f. as given in (8) and the same expectation
$\mathbb{E}[V_{i}^{(n)}]=(M_{p}(q),M_{p}(p))=(m_{p,q}^{q},1)$. Hence, the fact
that the expectation $m_{n,p,q}$ only converges to $m_{p,q}$ does not affect
our proofs. This is in keeping with classical results from large deviations
theory like the Theorem of Gärtner-Ellis (see [17, Theorem V.6]), where an
arbitrary (i.e. not necessarily i.i.d.) sequence of random variables is not
required to have a shared expectation, but rather that the sequence of the
(appropriately rescaled) c.g.f.s of the individual random variables in the
sequence converge to a fixed function with the origin in the interior of its
effective domain. The resulting LDP then considers deviation probabilities
from the limiting expectation as well. In the case of $\|\mathscr{Z}\|$ the
c.g.f.s of the $\mathscr{V}_{i}^{(n)}$ are not employed at all (neither
themselves nor their limit in $n$). Instead, we simply use the density
approximation in Proposition 5.1 for the sum of the $V_{i}^{(n)}$ and make use
of the independence of $U^{1/n}$ from the coordinates of $V_{i}^{(n)}$. Since
our main results assume $n\in\mathbb{N}$ to be sufficiently large (that is,
large enough for the local density approximations in Section 5 to hold), this
effectively means that for $n\in\mathbb{N}$ sufficiently large, the difference
of $m_{p,q}$ and $m_{n,p,q},\mathscr{m}_{n,p,q}$ is of order at most $o(1)$
and therefore does not affect our SLD estimates.

###  2.5. A few remarks on Weingarten maps and curvature

As outlined in the introduction, we will finish the proof of our first main
result in Theorem 3.1 by integrating over a previously established density
estimate via a result of Adriani and Baldi [1] for Laplacian integral
expansions. This result has a heavily geometric flavour and relies on the
Weingarten maps of certain hypersurfaces, which in our case are simply curves
in $\mathbb{R}^{2}$. We will therefore just give a brief reminder of the
Weingarten map in this setting, recall some of its properties, and refer to
the relevant literature (e.g. [25, 36]) or Adriani and Baldi [1] for a more
in-depth discussion of the topic.

In general, the Weingarten map of a smooth hypersurface
$M\subset\mathbb{R}^{d}$ at a point $p\in M$ is an endomorphism of the tangent
space $T_{p}M$ at $p$, mapping any $y\in T_{p}M$ to the directional derivative
of a normal field of $M$ in $p$ in the direction of $y$. However, as remarked
in [1, Example 4.3], for $d=2$, hypersurfaces simplify to planar curves and
the Weingarten map at a point $p$ simplifies to the absolute value of the
curvature $K(p)$ of the curve at $p$. For implicit curves, i.e. curves given
as the zero set of a function, we have the following formula for its curvature
from [20, Proposition 3.1]:

######  Lemma 2.9.

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

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

with derivatives $F_{[i,j]}=F_{[i,j]}(p)$ as in (1).

######  Remark 2.10.

  * i)

Given the set-up of the previous Lemma, straightforward calculation of the
above fraction gives that

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

  * ii)

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

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

##  3\. Main Results

Using the concepts and notation established in the previous section, we now
proceed to present our main results and their applications:

###  3.1. Sharp asymptotics for $q$-norms of random vectors in
${\mathbb{S}}_{p}^{n-1}$ and ${\mathbb{B}}_{p}^{n}$

For $Z^{(n)}\sim\mathbf{C}_{n,p}$, we want to give sharp asymptotics for the
probability $\mathbb{P}\left(n^{1/p-1/q}\,\|Z^{(n)}\|_{q}>z\right)$ for
$z>m_{p,q}$ such that $z^{*}\in\mathcal{J}_{p}$, with $z^{*}$ as defined in
Lemma 2.5. Before presenting our results, let us define the deviation-
dependent functions $\xi(z)$ and $\kappa(z)$, as mentioned also in the sharp
large deviation results of Bahadur and Ranga Rao [7]. For
$x\in\mathbb{R}^{2}$, we set

$\displaystyle\mathfrak{H}_{x}:=\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))$ (10)

to be the Hessian of the c.g.f. $\Lambda_{p}(\tau)$ in
$\tau\in\mathbb{R}^{2}$, evaluated at $\tau(x)$. For $z>m_{p,q}$ such that
$z^{*}\in\mathcal{J}_{p}$, we then define the deviation-dependent functions as

$\displaystyle\xi(z)^{2}:=\langle\mathfrak{H}_{z^{*}}\,\tau(z^{*}),\tau(z^{*})\rangle\,\det\mathfrak{H}_{z^{*}},$
(11)
$\displaystyle\kappa(z)^{2}:=1-\frac{\left(\tau(z^{*})_{1}^{2}+\tau(z^{*})_{2}^{2}\right)^{3/2}\,|pq(p-q)z^{q}|}{\big{|}\tau(z^{*})_{2}^{2}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{11}-2\tau(z^{*})_{1}\tau(z^{*})_{2}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{12}+\tau(z^{*})_{1}^{2}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{22}\big{|}\,(z^{2q}+p^{2}q^{-2})^{3/2}}.$
(12)

######  Theorem 3.1.

Let $1\leq q<p<\infty$, $n\in\mathbb{N}$, and $Z^{(n)}$ be a random vector in
${\mathbb{B}}^{n}_{p}$ with $Z^{(n)}\sim\mathbf{C}_{n,p}$. Then, for $n$
sufficiently large and any $z>m_{p,q}$ such that $z^{*}\in\mathcal{J}_{p}$, it
holds that

$\mathbb{P}\left(n^{1/p-1/q}\|Z^{(n)}\|_{q}>z\right)=\frac{1}{\sqrt{2\pi
n}\,\kappa(z)\xi(z)}\,e^{-n\,\Lambda_{p}^{*}(z^{*})}\,(1+o(1)).$

We want to do the same for
$\mathbb{P}\left(n^{1/p-1/q}\,\|\mathscr{Z}^{(n)}\|_{q}>z\right)$ with
$\mathscr{Z}^{(n)}\sim\mathbf{U}_{n,p}$ and $z>m_{p,q}$. Again, we start by
defining our deviation-dependent function for $z>m_{p,q}$

$\displaystyle\gamma(z)^{2}$ $\displaystyle:=$
$\displaystyle\det\mathfrak{H}_{z^{*}}\,\tau(z^{*})_{1}^{2}\,(qz^{q}\tau(z^{*})_{1}+1)^{2}\vphantom{\sum}$
(13)
$\displaystyle\times\,\Bigg{[}\frac{z^{2q}q^{2}}{p^{2}}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{11}+\frac{2z^{q}q}{p}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{12}+\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{22}+\,\tau(z^{*})_{1}\frac{z^{q}q(q-p)}{p^{2}}\Bigg{]}.$

######  Theorem 3.2.

Let $1\leq q<p<\infty$, $n\in\mathbb{N}$, and $\mathscr{Z}^{(n)}$ be a random
vector in ${\mathbb{B}}^{n}_{p}$ with $\mathscr{Z}^{(n)}\sim\mathbf{U}_{n,p}$.
Then, for $n$ sufficiently large and any $z>m_{p,q}$ such that
$z^{*}\in\mathcal{J}_{p}$, it holds that

$\mathbb{P}\left(n^{1/p-1/q}\|\mathscr{Z}^{(n)}\|_{q}>z\right)=\frac{1}{\sqrt{2\pi
n}\,\gamma(z)}\,e^{-n\,\Lambda_{p}^{*}(z^{*})}\,(1+o(1)).$

We have seen in Section 2.4 that $\|Z\|$ and $\|\mathscr{Z}\|$ both satisfy
LDPs with the same speed and rate function for $z>m_{p,q}$ such that
$z^{*}\in\mathcal{J}_{p}$, despite the underlying distributions being
different. Comparing Theorem 3.1 and Theorem 3.2 now paints a different
picture, with the sharp asymptotics for $\|Z\|$ and $\|\mathscr{Z}\|$ being
noticeably different. As mentioned in our introductory statements,
idiosyncratic phenomena of underlying distributions, which can be drowned out
on the LDP scale, are often still visible on the scale of sharp large
deviations. This is in keeping with what was shown in [37, Theorem 2.4,
Theorem 2.6] for one-dimensional projections of $\ell_{p}^{n}$-spheres and
$\ell_{p}^{n}$-balls.

######  Remark 3.3.

Let us draw a brief comparison between our results and the concentration
inequality that follows by the Gromov-Milman Theorem as discussed in [38,
Remark, p. 1062]. Therein, it is shown that the Gromov-Milman theorem from
[21] implies that for $1<q\leq p<\infty$ and a random vector
$Z^{(n)}\sim\mathbf{C}_{n,p}$, it holds that

$\mathbb{P}\Big{(}\big{|}n^{1/p-1/q}\|Z^{(n)}\|_{q}-m_{n,p,q}\big{|}\geq
z\Big{)}\leq C\,\exp\left(-c\,n\,z^{\max\\{2,p\\}}\right),$

where $C>0$ and $c>0$ are constants. If we consider the set-up of Theorem 3.1,
i.e. $1\leq q<p<\infty$ and $z>m_{n,p,q}$, and only consider deviations
without the absolute value, we can derive from the above that

$\mathbb{P}\Big{(}n^{1/p-1/q}\|Z^{(n)}\|_{q}>z\Big{)}\leq
C\,\exp\left(-c\,n\,z^{\max\\{2,p\\}}\right).\vspace{0.2cm}$

Comparing this with our sharp large deviation results from Theorem 3.1 for
$z>m_{p,q}$ such that $z^{*}\in\mathcal{J}_{p}$,

$\mathbb{P}\left(n^{1/p-1/q}\|Z^{(n)}\|_{q}>z\right)=\frac{1}{\sqrt{2\pi
n}\,\kappa(z)\xi(z)}\,e^{-n\,\Lambda_{p}^{*}(z^{*})}\,(1+o(1)),$

we can see that our results improve on the estimate in terms of
$n\in\mathbb{N}$ by a factor of $n^{-1/2}$ and give explicit and deviation-
dependent terms $\kappa(z)$ and $\xi(z)$ instead of fixed constants for all
deviations $z$.

######  Remark 3.4.

When comparing the SLD results in Theorem 3.1 and Theorem 3.2 to those of Liao
and Ramanan [37, Theorem 2.4, Theorem 2.6], one directly notices the core
difference in the settings. Liao and Ramanan examine projections of random
vectors on ${\mathbb{S}}^{n-1}_{p}$ and ${\mathbb{B}}^{n}_{p}$ with respective
distributions $\mathbf{C}_{n,p}$ and $\mathbf{U}_{n,p}$ onto fixed one-
dimensional subspaces, and therefore have to consider weighted sums of
dependent random vectors as probabilistic representations. Thus, all their
results have to be conditioned on the projection space and include additional
terms accounting for the specifics of the subspace. In our case however, the
probabilistic representations are given as sums of i.i.d. random variables
(see Section 4), which does not necessitate these additional factors.
Therefore, when using results from Liao and Ramanan [37], we adapt their usage
accordingly to the given probabilistic representations in our setting. Beyond
that however, the SLD results share several similarities, especially when
comparing the deviation-dependent terms $\kappa$, $\xi$ and $\gamma$, which
for $q=1$ are almost equal.

Both proofs of Theorem 3.1 and Theorem 3.2 contain three essential steps, as
already briefly mentioned in the introduction. The first will be rewriting the
probabilities in both theorems with respect to convenient probabilistic
representations, specifically $S^{(n)}$ and $\mathscr{S}^{(n)}$ given in (16)
of Section 4 as the respective empirical averages of the $V^{(n)}_{i}$ and
$\mathscr{V}^{(n)}_{i}$ in (6) and (7). The idea is to write the deviation
probabilities as an integral of their distribution over a given “deviation
area”. The second step is giving local density approximations for these
representations. Since the entries of both the $V^{(n)}$ and the
$\mathscr{V}^{(n)}$ are highly dependent, no canonical joint densities are
available to us to easily do so. However, their Fourier transforms can be
given explicitly, thus, for $n\in\mathbb{N}$ large enough one can use the
Fourier inversion theorem to write the densities of $S^{(n)}$ and
$\mathscr{S}^{(n)}$ as integrals of their Fourier transforms. Heuristically
speaking, this means that while the individual $V^{(n)}_{i}$ and
$\mathscr{V}^{(n)}_{i}$ do not possess densities in $\mathbb{R}^{2}$, but for
$n\in\mathbb{N}$ sufficiently large their empirical averages $S^{(n)}$ and
$\mathscr{S}^{(n)}$ asymptotically do. The resulting integrals can then be
approximated using the saddle point method. Since our representations are
given as sums of i.i.d. random vectors, for whom this has been done in
previous results (see, e.g., [10, 14, 42, 43]), we will not prove the density
approximations here explicitly. The third and final step then is to calculate
the integrals of these densities over their respective deviation area. For
$\|Z\|$, this is done by a result of Adriani and Baldi [1], which construes
the boundary of the deviation area and the level sets of the rate function in
the corresponding LDP as hypersurfaces, which are just planar curves in our
setting, and uses their Weingarten maps to approximate the integral. For
$\|\mathscr{Z}\|$, this is not applicable, as certain differentiability
conditions are no longer met. Thus, a result by Breitung and Hohenbichler [12]
is used, which allows for multi-dimensional Laplace integral approximations
under less restrictive differentiability conditions.

###  3.2. Intersection volumes of $\ell_{p}^{n}$-balls

We want to use our sharp large deviation results to further the findings of
Schechtman and Schmuckenschläger [44] and Schmuckenschläger [47] for
intersection volumes of $t$-multiples of different $\ell_{p}^{n}$-balls. We
will first give a brief overview of the original results. For
$p\in[1,\infty)$, we define
${\mathbb{D}}_{p}^{n}:=\mathrm{vol}_{n}({\mathbb{B}}_{p}^{n})^{-1/n}\,{\mathbb{B}}_{p}^{n}$
to be the volume normalized $\ell_{p}^{n}$-ball and recall that

$\mathrm{vol}_{n}({\mathbb{B}}_{p}^{n})=\frac{{\left(2\Gamma\left(1+\frac{1}{p}\right)\right)}^{n}}{\Gamma\left(1+\frac{n}{p}\right)}.$

We furthermore set

$c_{n,p}:=n^{1/p}\,\mathrm{vol}_{n}\left({\mathbb{B}}_{p}^{n}\right)^{1/n}\qquad\text{
and }\qquad c_{p}:=2\,e^{1/p}\,p^{1/p}\,\Gamma\left(1+\frac{1}{p}\right),$

and recall that it was shown in [44] that
$\lim\limits_{n\to\infty}c_{n,p}=c_{p}$. Moreover, for $p,q\in[1,\infty),p\neq
q$, we set

$c_{n,p,q}:=\frac{c_{n,p}}{c_{n,q}},\qquad\qquad
A_{n,p,q}:=\frac{c_{n,p}}{m_{p,q}\,c_{n,q}},\qquad\text{ and }\qquad
A_{p,q}:=\lim_{n\to\infty}A_{n,p,q}.$

Hence, it follows that

$A_{p,q}=\frac{c_{p}}{m_{p,q}\,c_{q}}=\frac{\Gamma\left(1+\frac{1}{p}\right)^{1+(1/q)}}{\Gamma\left(1+\frac{1}{q}\right)\Gamma\left(\frac{q+1}{p}\right)^{1/q}}\,\,e^{1/p-1/q}.$

Lastly, for $t\geq 0$ and $n\in\mathbb{N}$, we define $t_{n}\geq 0$ such that

$t_{n}\frac{A_{p,q}}{A_{n,p,q}}=t.$

Having established the necessary notation, we shall now recall the result of
Schmuckenschläger [47, Theorem 3.3]. Therein, it was shown that for
$p,q\in[1,\infty),p\neq q$, and $t\geq 0$ it holds that

$\mathrm{vol}_{n}\left({\mathbb{D}}_{p}^{n}\cap
t{\mathbb{D}}_{q}^{n}\right)\underset{n\to\infty}{\longrightarrow}\begin{cases}1&:A_{p,q}\,t>1\\\
\frac{1}{2}&:A_{p,q}\,t=1\\\ 0&:A_{p,q}\,t<1.\end{cases}$ (14)

To prove this, a central limit theorem for
$n^{1/p-1/q}\|\mathscr{Z}^{(n)}\|_{q}$ with
$\mathscr{Z}^{(n)}\sim\mathbf{U}_{n,p}$ and $p,q\in[1,\infty)$, $p\neq q$, is
shown in [47, Proposition 2.4, Proof of Theorem 3.2], since
$\mathrm{vol}_{n}({\mathbb{D}}_{p}^{n}\cap t{\mathbb{D}}_{q}^{n})$ can be
written as

$\displaystyle\mathrm{vol}_{n}({\mathbb{D}}_{p}^{n}\cap
t{\mathbb{D}}_{q}^{n})$ $\displaystyle=$
$\displaystyle\mathrm{vol}_{n}\left(\left\\{z\in{\mathbb{D}}_{p}^{n}:z\in
t_{n}\,\frac{A_{p,q}}{A_{n,p,q}}\,{\mathbb{D}}_{q}^{n}\right\\}\right)\vphantom{\int_{0}}$
(15) $\displaystyle=$
$\displaystyle\mathrm{vol}_{n}\left(\left\\{z\in{\mathbb{D}}_{p}^{n}:z\in
t_{n}\,A_{p,q}\,m_{p,q}\,\frac{c_{n,q}}{c_{n,p}}\,{\mathbb{D}}_{q}^{n}\right\\}\right)\vphantom{\int_{0}}$
$\displaystyle=$
$\displaystyle\mathrm{vol}_{n}\left(\left\\{z\in\mathrm{vol}_{n}({\mathbb{B}}_{p}^{n})^{-1/n}\,{\mathbb{B}}_{p}^{n}:z\in
t_{n}\,A_{p,q}\,m_{p,q}\,n^{1/q-1/p}\,\mathrm{vol}_{n}({\mathbb{B}}_{p}^{n})^{-1/n}\,{\mathbb{B}}_{q}^{n}\right\\}\right)\vphantom{\int_{0}}$
$\displaystyle=$
$\displaystyle\mathrm{vol}_{n}({\mathbb{B}}_{p}^{n})^{-1}\,\mathrm{vol}_{n}\left(\left\\{z\in{\mathbb{B}}_{p}^{n}:z\in
t_{n}\,A_{p,q}\,m_{p,q}\,n^{1/q-1/p}\,{\mathbb{B}}_{q}^{n}\right\\}\right)\vphantom{\int_{0}}$
$\displaystyle=$
$\displaystyle\mathbb{P}\left(n^{1/p-1/q}\|\mathscr{Z}^{(n)}\|_{q}\leq
t_{n}\,A_{p,q}\,m_{p,q}\right).$

However, we know from the Berry-Esseen Theorem (see [48, Theorem 2.1.3]) that
the error of the Gaussian approximation given by a central limit theorem
decreases with rate $n^{-1/2}$. Thus, using (15) and the central limit theorem
from [47], we can only infer a rate of convergence of $n^{-1/2}$ in (14).
Using Theorem 3.2, we can considerably refine that rate of convergence in the
first of the three cases in (14) from a sublinear rate to an exponential rate
for $1\leq q<p<\infty$.

######  Proposition 3.5.

Let $1\leq q<p<\infty$ and $n\in\mathbb{N}$. Using the notation established
above, it then holds for $t>m_{p,q}\,{c_{n,p,q}}^{-1}$ such that
$(t\,c_{n,p,q})^{*}\in\mathcal{J}_{p}$, and sufficiently large
$n\in\mathbb{N}$ that

$\mathrm{vol}_{n}\left({\mathbb{D}}_{p}^{n}\cap
t{\mathbb{D}}_{q}^{n}\right)=1-\frac{1}{\sqrt{2\pi
n}\,\gamma(t\,c_{n,p,q})}\,e^{-n\,\Lambda_{p}^{*}((t\,c_{n,p,q})^{*})}\,(1+o(1)).$

###### Proof.

Let $1\leq q<p<\infty$, $t>m_{p,q}\,{c_{n,p,q}}^{-1}$ such that
$(t\,c_{n,p,q})^{*}\in\mathcal{J}_{p}$, and assume $\mathscr{Z}^{(n)}$ is a
random vector in ${\mathbb{B}}_{p}^{n}$ with
$\mathscr{Z}^{(n)}\sim\mathbf{U}_{n,p}$. Using (15), we get that

$\mathrm{vol}_{n}\left({\mathbb{D}}_{p}^{n}\cap
t{\mathbb{D}}_{q}^{n}\right)=\mathbb{P}\left(n^{1/p-1/q}\|\mathscr{Z}^{(n)}\|_{q}\leq
t_{n}A_{p,q}\,m_{p,q}\right)=1-\mathbb{P}\left(n^{1/p-1/q}\|\mathscr{Z}^{(n)}\|_{q}>t_{n}A_{p,q}\,m_{p,q}\right).$

It now holds that, by $t>m_{p,q}\,{c_{n,p,q}}^{-1}$, we have that
$t\,m_{p,q}^{-1}\,c_{n,p,q}=t\,A_{n,p,q}=t_{n}\,A_{p,q}>1$, and hence
$t\,c_{n,p,q}=t_{n}\,A_{p,q}\,m_{p,q}>m_{p,q}$ with
$(t\,c_{n,p,q})^{*}\in\mathcal{J}_{p}$. Thus, by Theorem 3.2, it follows that

$\mathrm{vol}_{n}\left({\mathbb{D}}_{p}^{n}\cap
t{\mathbb{D}}_{q}^{n}\right)=1-\frac{1}{\sqrt{2\pi
n}\,\gamma(t\,c_{n,p,q})}\,e^{-n\,\Lambda_{p}^{*}((t\,c_{n,p,q})^{*})}\,(1+o(1)),$

which finishes our proof. ∎

###  3.3. One-dimensional projections of $\ell_{q}^{n}$-balls

In Remark 3.4 we have already discussed the differences between the setting of
the results of Liao and Ramanan [37] and the setting of this paper. However, a
geometrically similar result to those in [37] follows from Theorem 3.1. In
[28, Section 2.4 ] Kabluchko, Prochno and Thäle derived a central limit
theorem for the the length of the projection of an $\ell_{p}^{n}$-ball onto
the line spanned by a random vector $\theta^{(n)}\in{\mathbb{S}}^{n-1}$ with
$\theta^{(n)}\sim\mathbf{C}_{n,2}$ as a corollary of their main results. We
will proceed similarly and derive sharp large deviation results in the same
setting. To be specific, in [37] sharp asymptotics where provided for the
scalar product of a random vector $Z^{(n)}\sim\mathbf{C}_{n,p}$ on
${\mathbb{S}}^{n-1}_{p}$ with a random vector
$\theta^{(n)}\sim\mathbf{C}_{n,2}$ on ${\mathbb{S}}^{n-1}$, which can be
negative. We, on the other hand, consider the absolute value of the scalar
product of such random vectors, thereby only considering non-negative values.

In the following, for $q\in[1,\infty]$, define its conjugate $q^{*}$ via
${1}/{q}+{1}/{q^{*}}=1$, setting ${1}/{\infty}=0$ by convention. Furthermore,
for a vector $\theta^{(n)}\in{\mathbb{S}}^{n-1}$, we write
$P_{\theta^{(n)}}{\mathbb{B}}_{q}^{n}$ for the projection of
${\mathbb{B}}_{q}^{n}$ onto the line spanned by $\theta^{(n)}$. Then, our
quantity of interest is the projection length
$\mathrm{vol}_{1}\left(P_{\theta^{(n)}}{\mathbb{B}}_{q}^{n}\right)$.

######  Corollary 3.6.

Let $2<q\leq\infty$ and $\theta^{(n)}\in{\mathbb{S}}^{n-1}$ be a random vector
with $\theta^{(n)}\sim\mathbf{C}_{n,2}$. Then, for any $z>2\,m_{2,q^{*}}$ such
that $(\frac{z}{2})^{*}\in\mathcal{J}_{p}$, and sufficiently large
$n\in\mathbb{N}$, it holds that

$\displaystyle\mathbb{P}\left(n^{1/2-1/q}\,\mathrm{vol}_{1}\left(P_{\theta^{(n)}}{\mathbb{B}}_{q}^{n}\right)>z\right)=\frac{1}{\sqrt{2\pi
n}\,\kappa\left(\frac{z}{2}\right)\xi\left(\frac{z}{2}\right)}\,e^{-n\,\Lambda^{*}_{2}({(\frac{z}{2})}^{*})}\,(1+o(1)),$

with $\Lambda_{2}$ as in (8) and $\xi,\kappa$ as in (11), (12), respectively,
for $q^{*}$ and $p=2$.

###### Proof.

It holds that

$\mathbb{P}\left(n^{1/2-1/q}\,\mathrm{vol}_{1}\left(P_{\theta^{(n)}}{\mathbb{B}}_{q}^{n}\right)>z\right)=\mathbb{P}\left(n^{1/2-1/q}\,2\sup_{x\in{\mathbb{B}}_{q}^{n}}|\langle
x,\theta^{(n)}\rangle|>z\right)=\mathbb{P}\left(n^{1/2-1/q}\,\|\theta^{(n)}\|_{q^{*}}>\frac{z}{2}\right).$

Since $2<q\leq\infty$, we have $1\leq q^{*}<2=p$, whereby we can apply Theorem
3.1 to the above to get that

$\mathbb{P}\left(n^{1/2-1/q}\,\mathrm{vol}_{1}\left(P_{\theta^{(n)}}{\mathbb{B}}_{q}^{n}\right)>z\right)=\frac{1}{\sqrt{2\pi
n}\,\kappa\left(\frac{z}{2}\right)\xi\left(\frac{z}{2}\right)}\,e^{-n\,\Lambda^{*}_{2}\left((\frac{z}{2})^{*}\right)}\,(1+o(1)),$

with $\Lambda_{2},\xi,\kappa$ as described above, which concludes our proof. ∎

##  4\. Probabilistic Representation

Recalling the definitions of the random vectors $V^{(n)}$ and
$\mathscr{V}^{(n)}$ from (6) and (7), we define

$S^{(n)}:=\frac{1}{n}\sum_{i=1}^{n}V^{(n)}_{i}\qquad\text{ and
}\qquad\mathscr{S}^{(n)}:=\frac{1}{n}\sum_{i=1}^{n}\mathscr{V}^{(n)}_{i}$ (16)

as the empirical averages of their respective coordinates. Furthermore, we
define the sets

$D_{z}:=\\{(t_{1},t_{2})\in\mathbb{R}^{2}:t_{1},t_{2}>0,t_{1}^{1/q}\,t_{2}^{-1/p}>z\\},$

and

$\mathscr{D}_{z}:=\\{(t_{1},t_{2},t_{3})\in\mathbb{R}^{3}:t_{1},t_{2}>0,\,t_{3}\in(0,1],\,t_{3}\,t_{1}^{1/q}\,t_{2}^{-1/p}>z\\}.$

It then follows from the reformulations of $\|Z^{(n)}\|_{q}$ and
$\|\mathscr{Z}^{(n)}\|_{q}$ in (4) and (5) that we can write the probabilities
within Theorem 3.1 and Theorem 3.2 with respect to $S^{(n)}$ and
$\mathscr{S}^{(n)}$, respectively, as

$\mathbb{P}\left(n^{1/p-1/q}\,\|Z^{(n)}\|_{q}>z\right)=\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{n}|Y^{(n)}_{i}|^{q}>z^{q}\,{\left(\frac{1}{n}\sum_{i=1}^{n}|Y^{(n)}_{i}|^{p}\right)}^{\frac{q}{p}}\right)=\mathbb{P}\left(S^{(n)}\in
D_{z}\right),$ (17)

and

$\mathbb{P}\left(n^{1/p-1/q}\,\|\mathscr{Z}^{(n)}\|_{q}>z\right)=\mathbb{P}\left(U^{\frac{q}{n}}\,\frac{1}{n}\sum_{i=1}^{n}|Y^{(n)}_{i}|^{q}>z^{q}\,{\left(\frac{1}{n}\sum_{i=1}^{n}|Y^{(n)}_{i}|^{p}\right)}^{\frac{q}{p}}\right)=\mathbb{P}\left(\mathscr{S}^{(n)}\in\mathscr{D}_{z}\right).$
(18)

We refer to these sets as “deviation areas”, as $S^{(n)}$ or
$\mathscr{S}^{(n)}$ lying in $D_{z}$ or $\mathscr{D}_{z}$ represents a
deviation of $\|Z^{(n)}\|_{q}$ and $\|\mathscr{Z}^{(n)}\|_{q}$. Note that the
boundaries of the deviation areas

$\partial
D_{z}=\\{(t_{1},t_{2})\in\mathbb{R}^{2}:t_{1},t_{2}>0,t_{1}^{1/q}\,t_{2}^{-1/p}=z\\}$

and

$\partial\mathscr{D}_{z}=\\{(t_{1},t_{2},t_{3})\in\mathbb{R}^{3}:t_{1},t_{2}>0,\,t_{3}\in(0,1],\,t_{3}\,t_{1}^{1/q}\,t_{2}^{-1/p}=z\\}$

are the same sets given by the infimum conditions in the respective LDPs for
$\|Z\|$ and $\|\mathscr{Z}\|$ in Proposition 2.4 and Proposition 2.6. The fact
that for $z>m_{p,q}$, the rate functions of these LDPs both assume a unique
minimum on $\partial D_{z}$ and $\partial\mathscr{D}_{z}$, respectively, as
was shown in Lemma 2.5 and Lemma 2.7, will be essential to the proof of our
main results in Sections 6 and 7. We can expand this unique infimum property
onto the entirety of $\overline{D_{z}}$ and $\overline{\mathscr{D}_{z}}$, as
the following lemma will show:

######  Lemma 4.1.

Assume the same set-up as in Lemma 2.5 and Lemma 2.7. Let $z>m_{p,q}$ such
that $z^{*}\in\mathcal{J}_{p}$. Then

  1. i)

$z^{*}=(z^{q},1)$ is the unique point at which $\Lambda_{p}^{*}$ attains its
infimum on $\overline{D_{z}}$,

  2. ii)

$z^{**}=(z^{q},1,1)$ is the unique point at which $\mathcal{I}_{\mathscr{S}}$
attains its infimum on $\overline{\mathscr{D}_{z}}$.

###### Proof.

We start off by showing i). Let $t\in\mathbb{R}^{2}$ such that $t\in
D_{z}^{\circ}$, meaning $t_{1}^{1/q}t_{2}^{-1/p}>z$. Then, for
$\tilde{z}:=t_{1}^{1/q}t_{2}^{-1/p}$ we assume that
$\tilde{z}\in\mathcal{J}_{p}$, as otherwise our claim trivially holds by
Remark 2.3. We then have that $t\in\partial D_{\tilde{z}}$, thus, by Lemma
2.5,
$\Lambda_{p}^{*}(t_{1},t_{2})>\Lambda_{p}^{*}(\tilde{z}^{q},1)=\mathcal{I}_{\|Z\|}(\tilde{z})$.
We know that $\Lambda_{p}^{*}$ is a convex function with a root in the
expectation $(m_{p,q}^{q},1)$ of the $V_{i}^{(n)}$ from (6), since it is a
rate function (apply arguments from [16, Lemma 2.2.5] in $\mathbb{R}^{2}$). We
also show in Lemma 6.2 $ii)$ that
$\mathcal{H}_{x}\Lambda_{p}^{*}(x)=\mathfrak{H}_{x}^{-1}$, and have argued in
Section 2 for why $\mathfrak{H}_{x}$ is positive definite on
$\mathcal{D}_{p}$, hence the Hessian of $\Lambda_{p}^{*}$ is also positive
definite on $\mathcal{J}_{p}$, giving us the strict convexity of
$\Lambda_{p}^{*}$ and, thereby strict convexity of
$\mathcal{I}_{\|Z\|}(z)=\Lambda_{p}^{*}(z^{*})$ on $\mathcal{J}_{p}$. Hence we
know that $\mathcal{I}_{\|Z\|}(z)$ is strictly increasing in $z$ for
$z>m_{p,q}$. Thus, as $\tilde{z}>z>m_{p,q}$, it follows that

$\Lambda_{p}^{*}(t_{1},t_{2})>\Lambda_{p}(\tilde{z}^{q},1)=\mathcal{I}_{\|Z\|}(\tilde{z})>\mathcal{I}_{\|Z\|}(z)=\Lambda_{p}^{*}(z^{q},1)=\Lambda_{p}^{*}(z^{*}),$

showing that $z^{*}=(z^{q},1)$ minimizes $\Lambda_{p}^{*}$ over
$\overline{D_{z}}$. The proof of ii) is analogous, also using the strict
monotonicity of the rate function. ∎

Suppose that the distributions of $S^{(n)}$ and $\mathscr{S}^{(n)}$ have
respective densities $h^{(n)}$ and $\mathscr{h}^{(n)}$. Then we can formulate
our probabilities of interest as

$\mathbb{P}\left(n^{1/p-1/q}\,\|Z^{(n)}\|_{q}>z\right)=\mathbb{P}\left(S^{(n)}\in
D_{z}\right)=\int_{D_{z}}h^{(n)}(x)\,\textup{d}x,$ (19)

and

$\mathbb{P}\left(n^{1/p-1/q}\,\|\mathscr{Z}^{(n)}\|_{q}>z\right)=\mathbb{P}\left(\mathscr{S}^{(n)}\in\mathscr{D}_{z}\right)=\int_{\mathscr{D}_{z}}\mathscr{h}^{(n)}(x)\,\textup{d}x.$
(20)

The following section will be devoted to showing the existence of these
densities $h^{(n)}$ and $\mathscr{h}^{(n)}$ and presenting them explicitly,
while Sections 6 and 7 will then approximate their integrals over their
respective deviation areas $D_{z}$ and $\mathscr{D}_{z}$.

##  5\. Joint Density Estimate

Recalling the notation and definitions established in Section 2, we assume the
same set-up as in Section 4 and can formulate the following local limit
theorems for the densities $h^{(n)}$ and $\mathscr{h}^{(n)}$ of our
probabilistic representations $S^{(n)}$ and $\mathscr{S}^{(n)}$.

######  Proposition 5.1.

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

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

where $\mathfrak{H}_{x}:=\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))$ as in (10).

For the proof of this, we refer to the results of Borovkov and Rogozin [10] or
their convenient reformulation in [1, Theorem 3.1]. Therein, a local density
estimate is derived for a sum of i.i.d. random vectors in $\mathbb{R}^{d}$ via
the saddle point method. As discussed in Section 2.2, this means, one writes
the density via the Fourier inversion theorem as a complex integral over its
Fourier transform and then uses Cauchy’s theorem to deform the path of
integration, such that it passes through a complex saddle point. For
sufficiently large $n\in\mathbb{N}$, the mass of the integral then heavily
concentrates around that saddle point and standard integral expansion methods
can be used to great effect. Naturally, this requires the conditions of the
Fourier inversion theorem to be met, that is, the Fourier transform of the
density has to be integrable. In [1, Theorem 3.1] this follows from the
assumption that all the i.i.d. random vectors have a common bounded density,
though it is noted in [1, Remark 3.2], that this can be replaced by any
argument ensuring that the Fourier inversion theorem can be applied. In our
setting, the i.i.d. vectors are given by
$V^{(n)}_{i}:=({|Y_{i}^{(n)}|}^{q},{|Y_{i}^{(n)}|}^{p})$, whose coordinates
are highly dependent, thus such a density of the $V^{(n)}_{i}$ is not
available. However, one can write the Fourier transform of $V^{(n)}_{i}$ with
respect to the underlying distribution $\mathbf{N}_{p}$ of the $Y_{i}^{(n)}$,
and then infer integrability via the properties of its density $f_{p}$ and the
Hausdorff-Young inequality, as was done by Liao and Ramanan in [37, Lemma
6.1]. As the considered settings are quite similar, virtually the same
arguments can be applied in our case, thereby making sure our referral to [1,
Theorem 3.1] is indeed justified.

######  Proposition 5.2.

For $\mathscr{S}^{(n)}=\frac{1}{n}\sum_{i=1}^{n}\mathscr{V}^{(n)}_{i}$ with
$\mathscr{V}^{(n)}_{i}=({|Y^{(n)}_{i}|}^{q},{|Y^{(n)}_{i}|}^{p},U^{1/n}),\,Y^{(n)}_{i}\sim\mathbf{N}_{p}$
i.i.d., $U$ uniformly distributed on $[0,1]$ independently of the
$Y_{i}^{(n)}$, and $x=(x_{1},x_{2})\in\mathcal{J}_{p}$, $y\in(0,1]$, it holds
that for sufficiently large $n\in\mathbb{N}$ the distribution of
$\mathscr{S}^{(n)}$ has Lebesgue density

$\mathscr{h}^{(n)}(x_{1},x_{2},y)=\frac{n^{2}}{2\pi}\,y^{-1}{(\det\mathfrak{H}_{x})}^{-1/2}\,e^{-n\,\mathcal{I}_{\mathscr{S}}(x_{1},x_{2},y)}\,(1+o(1)),$

where $\mathcal{I}_{\mathscr{S}}(x_{1},x_{2},y):=[\Lambda_{p}^{*}(x)-\log(y)]$
and $\mathfrak{H}_{x}:=\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))$ as in (10).

###### Proof.

By direct calculation, we can see for $y\in[0,1]$ that
$\mathbb{P}\left(U^{1/n}\leq y\right)=\mathbb{P}\left(U\leq
y^{n}\right)=y^{n}$, giving that the density of $U^{1/n}$ is given by
$f_{U^{1/n}}(y)=n\,y^{n-1}$. As $U^{1/n}$ is independent of the $Y_{i}^{(n)}$,
and thereby also of $S^{(n)}=({|Y^{(n)}_{i}|}^{q},{|Y^{(n)}_{i}|}^{p})$, the
density of
$\mathscr{S}^{(n)}=\frac{1}{n}\sum_{i=1}^{n}({|Y^{(n)}_{i}|}^{q},{|Y^{(n)}_{i}|}^{p},U^{1/n})$
is given by the product of their densities, hence

$\mathscr{h}^{(n)}(x_{1},x_{2},y)=h^{(n)}(x_{1},x_{2})f_{U^{1/n}}(y)=\frac{n^{2}}{2\pi}\,y^{-1}{(\det\mathfrak{H}_{x})}^{-1/2}\,e^{-n\,[\Lambda_{p}^{*}(x)-\log(y)]}\,(1+o(1)).$

This completes our proof. ∎

##  6\. Proof of the Main Result for $\ell_{p}^{n}$-spheres

In (19) we have reformulated the deviation probability
$\mathbb{P}\left(n^{1/p-1/q}\,\|Z^{(n)}\|_{q}>z\right)$ as an integral of the
density estimate $h^{(n)}$ of the probabilistic representation $S^{(n)}$ over
the deviation area $D_{z}$. In Proposition 5.1 we have then given $h^{(n)}$
explicitly. For the proof of Theorem 3.1 it remains to calculate that
integral. To do so, the integral will be split up into a neighbourhood $B_{z}$
of the point $z^{*}$, that has been shown in Lemma 4.1 to be the infimum of
$\Lambda_{p}^{*}$ over $\bar{D}_{z}$, and its complement $B_{z}^{c}$. The LDP
from Proposition 2.4 will be used to show the negligibility of the integral
outside of the neighbourhood of $z^{*}$. Within the neighbourhood $B_{z}$, we
use a result from Adriani and Baldi [1], which uses the Weingarten maps of the
planar curves given by the boundary of $D_{z}\cap B_{z}$ and the level set of
$\Lambda_{p}^{*}$ at $z^{*}$, to compute the integral. Following that, we will
give these Weingarten maps explicitly, finishing our proof.

###### Proof of Theorem 3.1.

We assume the set-up of Theorem 3.1 and use the reformulation (19) to proceed
by considering $\mathbb{P}\left(S^{(n)}\in D_{z}\right)$. Let
$B_{z}\subset\mathbb{R}^{2}$ be an open neighbourhood around $z^{*}$, small
enough that $B_{z}\subset\mathcal{J}_{p}$. Then it holds that

$\mathbb{P}(S^{(n)}\in
D_{z})=\int_{D_{z}}h^{(n)}(x)\,\textup{d}x=\int_{D_{z}\cap\,B_{z}}h^{(n)}(x)\,\textup{d}x+\int_{D_{z}\cap\,B_{z}^{c}}h^{(n)}(x)\,\textup{d}x.$
(21)

Since $z^{*}\notin B_{z}^{c}$, by Lemma 4.1 , there exists an $\eta>0$, such
that

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

and thus, by the LDP in Proposition 2.4, it holds that

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

This gives us that

$\mathbb{P}\left(S^{(n)}\in D_{z}\cap\,B_{z}^{c}\right)\leq
e^{-n\,\Lambda_{p}^{*}(z^{*})-n\,\eta}\,(1+o(1))=\frac{1}{e^{n\,\eta}}\,e^{-n\Lambda_{p}^{*}(z^{*})}(1+o(1)).$
(22)

Furthermore, by our density estimate in Proposition 5.1, it holds that

$\int_{D_{z}\cap\,B_{z}}h^{(n)}(x)\,\textup{d}x=\frac{n}{2\pi}\,\int_{D_{z}\cap\,B_{z}}{(\det\mathfrak{H}_{x})}^{-1/2}\,e^{-n\,\Lambda_{p}^{*}(x)}\,\textup{d}x(1+o(1)).$
(23)

To calculate this explicitly, we will rely on a technique established in [1,
Proof of Theorem 4.4]. Therein, an asymptotic integral expansion of Bleistein
and Handelsmann [9, Equation (8.3.63)] for Laplace integrals is reformulated
via the Weingarten maps of the integration area and the level set of the
exponential function at its minimum, both seen as hypersurfaces. We will
present it as one concise result, similar to that formulated in [37, Lemma
4.6].

######  Proposition 6.1.

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

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

and denote by $L_{D}$ and $L_{\phi}$ their respective Weingarten maps at
$x^{*}$. Then, for sufficiently large $n\in\mathbb{N}$, it holds that

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

The proof of this is given by first applying the result from [9, Equation
(8.3.63)] for Laplace-type integrals and then using the reformulation of the
terms therein from [1, Equation (4.6)] with respect to the Weingarten map.

Let us now check that the above conditions hold for the integral in (23). We
have that $D_{z}\cap\,B_{z}$ is bounded, and for $z>m_{p,q}$, we can write
$\partial D_{z}$ as the graph of the infinitely differentiable function
$f:(0,\infty)\to(0,\infty)$ with $f(t_{1})=z^{-p}t_{1}^{p/q}$, thus both
$\partial D_{z}$ and $\partial(D_{z}\cap\,B_{z})$ are differentiable planar
curves. As discussed in Section 2, it holds for $x\in\mathcal{J}_{p}$ that
$g_{x}(\tau):=\langle x,\tau\rangle-\Lambda_{p}(\tau)$ has a unique supremum
$\tau(x)$, i.e. $x-\nabla_{\tau}\Lambda_{p}(\tau)=0$ has a unique solution in
$(x,\tau)$. It was also dicussed that $\mathcal{H}_{\tau}\Lambda_{p}(\tau)$ is
invertible for all $\tau\in\mathcal{D}_{p}$, thus, it follows from the
implicit function theorem that $x\mapsto\tau(x)$ is as differentiable in $x$
as $(x,\tau)\mapsto(x-\nabla_{\tau}\Lambda_{p}(\tau))$ is in $\tau$. As
$\Lambda_{p}$ is the logarithm of the joint m.g.f. of the $V_{i}^{(n)}$ from
(6), the components of its derivatives are themselves infinitely
differentiable within $\mathcal{D}_{p}$ by the standard properties of the
m.g.f. (see e.g. [24, Theorem 5.4]), yielding that $\tau(x)$ is infinitely
differentiable on $\mathcal{J}_{p}$. This overall gives us the infinite
differentiability of $\Lambda_{p}(\tau(x))$ and thereby also of
$\Lambda_{p}^{*}(x)$ on $\mathcal{J}_{p}$. For $B_{z}$ chosen small enough, it
then follows for any $z>m_{p,q}$ with $z^{*}\in\mathcal{J}_{p}$ that
$\Lambda_{p}^{*}$ is twice differentiable on $D_{z}\cap\,B_{z}$. Nonnegativity
of $\Lambda_{p}^{*}$ follows directly by the standard properties of rate
functions (apply arguments from e.g. [16, Lemma 2.2.5] in $\mathbb{R}^{2}$).
By the infinite differentiability of $\Lambda_{p}(\tau(x))$ in $x$, we get the
differentiability of
$g(x):={(\det\mathfrak{H}_{x})}^{-1/2}={(\det\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))}^{-1/2}$
in $x$. Lemma 4.1 gives us the uniqueness of
$z^{*}=(z^{q},1)\,\in\partial(D_{z}\cap\,B_{z})$ as an infimum on
$\overline{D}_{z}$ and $\overline{D_{z}\cap\,B_{z}}$.

Thus, in view of the above, we can use Proposition 6.1 for
$D=D_{z}\cap\,B_{z}\subset\mathbb{R}^{2}$ with
$g(x)={(\det\mathfrak{H}_{x})}^{-1/2}$, $\phi(x)=\Lambda_{p}^{*}(x)$, and
$x^{*}=z^{*}$, and get that

$\displaystyle\displaystyle\int_{D_{z}\cap\,B_{z}}h^{(n)}(x)\,\textup{d}x$
$\displaystyle=\displaystyle\frac{n}{2\pi}\,\frac{{(2\pi)}^{1/2}\,\det(L_{\Lambda}^{-1}(L_{\Lambda}-L_{D}))^{-1/2}\,(\det\mathfrak{H}_{z^{*}})^{-1/2}\,e^{-n\,\Lambda_{p}^{*}(z^{*})}}{n^{3/2}\,\langle{\mathcal{H}_{x}\,\Lambda_{p}^{*}(z^{*})}^{-1}\,\nabla_{x}\,\Lambda_{p}^{*}(z^{*}),\nabla_{x}\,\Lambda_{p}^{*}(z^{*})\rangle^{1/2}}\,(1+o(1)),\quad$
(24)

for the respective Weingarten maps at $z^{*}$ of the curves

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

Let us present the following identities for some of the terms in the fraction
above, resulting from the definition of $\tau(x)$ and the properties of the
Legendre-Fenchel transform:

######  Lemma 6.2.

It holds that

  1. i)

$\nabla_{x}\Lambda_{p}^{*}(x)=\tau(x),$

  2. ii)

$\mathcal{H}_{x}\Lambda_{p}^{*}(x)={\mathfrak{H}_{x}}^{-1}.$

###### Proof.

We start by showing that $\nabla_{x}\,\Lambda_{p}^{*}(x)=\tau(x)$. We have
defined $\tau(x)$ as the supremum of $[\langle
x,\tau\rangle-\Lambda_{p}(\tau)]$ in $\tau\in\mathbb{R}^{2}$ (see (9)), thus
it follows that

$\nabla_{\tau}\big{[}\langle
x,\tau\rangle-\Lambda_{p}(\tau)\big{]}\big{|}_{\tau=\tau(x)}=x-\nabla_{\tau}\Lambda_{p}(\tau(x))=0.$
(25)

With this, it follows that

$\displaystyle\nabla_{x}\Lambda_{p}^{*}(x)$ $\displaystyle=$
$\displaystyle\nabla_{x}\big{[}\langle
x,\tau(x)\rangle-\Lambda_{p}(\tau(x))\big{]}\vphantom{\sum}$ $\displaystyle=$
$\displaystyle\tau(x)+J_{x}\tau(x)\,x-\nabla_{x}\Lambda_{p}(\tau(x))\vphantom{\sum}$
$\displaystyle=$
$\displaystyle\tau(x)+J_{x}\tau(x)\,x-J_{x}\tau(x)\nabla_{\tau}\Lambda_{p}(\tau(x))\vphantom{\sum}$
$\displaystyle=$
$\displaystyle\tau(x)+J_{x}\tau(x)\,\big{[}x-\nabla_{\tau}\Lambda_{p}(\tau(x))\big{]}\vphantom{\sum}$
$\displaystyle=$ $\displaystyle\tau(x).\vphantom{\sum}$

Let us now prove that
$\mathcal{H}_{x}\Lambda_{p}^{*}(x)={\mathfrak{H}_{x}}^{-1}$. On the one hand,
it follows from the above that

$\mathcal{H}_{x}\Lambda_{p}^{*}(x)=J_{x}\tau(x),$ (26)

while on the other hand, it holds that

$\displaystyle\mathcal{H}_{x}\Lambda_{p}^{*}(x)$ $\displaystyle=$
$\displaystyle\mathcal{H}_{x}\big{[}\langle
x,\tau(x)\rangle-\Lambda_{p}(\tau(x))\big{]}\vphantom{\sum}$ (27)
$\displaystyle=$ $\displaystyle\mathcal{H}_{x}\big{[}\langle
x,\tau(x)\rangle\big{]}-\mathcal{H}_{x}\big{[}\Lambda_{p}(\tau(x))\big{]}\vphantom{\sum}$
$\displaystyle=$ $\displaystyle J_{x}\big{[}\nabla_{x}\langle
x,\tau(x)\rangle\big{]}-J_{x}\big{[}\nabla_{x}\Lambda_{p}(\tau(x))\big{]}\vphantom{\sum}$
$\displaystyle=$ $\displaystyle
J_{x}\big{[}\tau(x)+J_{x}\tau(x)x\big{]}-J_{x}\big{[}J_{x}\tau(x)\,\nabla_{\tau}\Lambda_{p}(\tau(x))\big{]}\vphantom{\sum}$
$\displaystyle=$ $\displaystyle
J_{x}\tau(x)+J_{x}\big{[}J_{x}\tau(x)x\big{]}-\mathcal{H}_{x}\tau(x)\,\nabla_{\tau}\Lambda_{p}(\tau(x))-J_{x}\tau(x)J_{x}\big{[}\nabla_{\tau}\Lambda_{p}(\tau(x))\big{]}\vphantom{\sum}$
$\displaystyle=$ $\displaystyle
2J_{x}\tau(x)+\mathcal{H}_{x}\tau(x)\big{[}x-\nabla_{\tau}\Lambda_{p}(\tau(x))\big{]}-J_{x}\tau(x)\,J_{x}\tau(x)\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))\vphantom{\sum}$
$\displaystyle=$ $\displaystyle
2J_{x}\tau(x)-J_{x}\tau(x)\,J_{x}\tau(x)\,\mathcal{H}_{\tau}\Lambda_{p}(\tau(x)).\vphantom{\sum}$

Equating the terms (26) and (27) yields

$\begin{array}[]{crcl}&J_{x}\tau(x)&=&2J_{x}\tau(x)-J_{x}\tau(x)\,J_{x}\tau(x)\,\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))\vphantom{\sum\limits_{0}}\\\
\Leftrightarrow&0&=&J_{x}\tau(x)-J_{x}\tau(x)\,J_{x}\tau(x)\,\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))\vphantom{\sum\limits_{0}}\\\
\Leftrightarrow&0&=&I_{2}-J_{x}\tau(x)\,\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))\vphantom{\sum\limits_{0}}\\\
\Leftrightarrow&J_{x}\tau(x)&=&\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))^{-1},\\\
\end{array}$

where $I_{2}$ denotes the identity matrix in $\mathbb{R}^{2}$. Again using
(26) on the above yields

$\mathcal{H}_{x}\Lambda_{p}^{*}(x)=J_{x}\tau(x)=\mathcal{H}_{\tau}\Lambda_{p}(\tau(x))^{-1}=\mathfrak{H}_{x}^{-1},$

and thereby finishes the proof. ∎

Via Lemma 6.2, we get

$\Big{\langle}{\mathcal{H}_{x}\,\Lambda_{p}^{*}(z^{*})}^{-1}\,\nabla_{x}\,\Lambda_{p}^{*}(z^{*}),\nabla_{x}\,\Lambda_{p}^{*}(z^{*})\Big{\rangle}=\Big{\langle}\mathfrak{H}_{z^{*}}\,\tau(z^{*}),\tau(z^{*})\Big{\rangle}.$

With the definition of $\displaystyle\xi(z)^{2}$ in (11) the integral in (24)
hence simplifies as follows:

$\displaystyle\displaystyle\int_{D_{z}\cap\,B_{z}}h^{(n)}(x)\,\textup{d}x$
$\displaystyle=\displaystyle\frac{1}{\sqrt{2\pi
n}\,\xi(z)}\,{\big{(}\det(L_{\Lambda}^{-1}(L_{\Lambda}-L_{D})\big{)}}^{-1/2}\,e^{-n\,\Lambda_{p}^{*}(z^{*})}\,(1+o(1)).$
(28)

We see that it only remains to prove that
$\det(L_{\Lambda}^{-1}(L_{\Lambda}-L_{D}))=\kappa(z)^{2}$. We proceed to
calculate the Weingarten maps of the curves $\mathscr{C}_{D}$ and
$\mathscr{C}_{\Lambda}$ explicitly. As discussed in Section 2.5, the
Weingarten map of a planar curve at a point $x$ reduces to the absolute value
of its curvature in $x$. As previously mentioned, $\partial D_{z}$ is the
graph of a function $f:(0,\infty)\to(0,\infty)$ with
$f(t_{1})=z^{-p}t_{1}^{p/q}$. Thus, the same holds locally for
$\mathscr{C}_{D}=\partial(D_{z}\cap\,B_{z})$ in a neighbourhood of $z^{*}$, so
by the curvature formula for graphs of functions, as seen in Remark 2.10 ii),
it holds that

$L_{D}=\frac{|f^{\prime\prime}(z^{q})|}{(1+f^{\prime}(z^{q})^{2})^{3/2}},$

where

$f^{\prime}(t_{1})^{2}={\left(pq^{-1}\,z^{-p}\,{t_{1}}^{(p/q)-1}\right)}^{2}\Rightarrow
f^{\prime}(z^{q})^{2}=p^{2}q^{-2}z^{-2q},$

and

$f^{\prime\prime}(t_{1})=pq^{-1}\left(pq^{-1}-1\right)z^{-p}\,{t_{1}}^{(p/q)-2}\Rightarrow
f^{\prime\prime}(z^{q})=(p^{2}-pq)q^{-2}\,z^{-2q}.$

This yields

$L_{D}=\frac{|(p^{2}-pq)q^{-2}\,z^{-2q}|}{(1+p^{2}q^{-2}z^{-2q})^{3/2}}=\frac{|pq(p-q)z^{q}|}{(z^{2q}+p^{2}q^{-2})^{3/2}}.$
(29)

The curve $\mathscr{C}_{\Lambda}$ is the zero set of the function
$F(x):=\Lambda_{p}^{*}(x)-\Lambda_{p}^{*}(z^{*})$. From Lemma 6.2 we know that

$\displaystyle(F_{[1,0]},F_{[0,1]})=\Big{(}\frac{\partial}{\partial
x_{1}}\Lambda_{p}^{*}(z^{*}),\,\frac{\partial}{\partial
x_{2}}\Lambda_{p}^{*}(z^{*})\Big{)}=\tau(z^{*})$

and

$\left(\begin{array}[]{rrr}F_{[2,0]}&F_{[1,1]}\\\ F_{[1,1]}&F_{[0,2]}\\\
\end{array}\right)=\displaystyle\left(\begin{array}[]{rrr}\displaystyle\frac{\partial^{2}}{\partial^{2}x_{1}}\Lambda_{p}^{*}(z^{*})&\displaystyle\frac{\partial^{2}}{\partial
x_{2}\partial x_{1}}\Lambda_{p}^{*}(z^{*})\\\ \\\
\displaystyle\frac{\partial^{2}}{\partial x_{1}\partial
x_{2}}\Lambda_{p}^{*}(z^{*})&\displaystyle\frac{\partial^{2}}{\partial^{2}x_{2}}\Lambda_{p}^{*}(z^{*})\\\
\end{array}\right)=\mathfrak{H}_{z^{*}}^{-1},$

for derivatives $F_{[i,j]}=F_{[i,j]}(z^{*})$ as in (1). Hence, by the
curvature formula for implicit curves from Lemma 2.9 and Remark 2.10 i), we
get

$L_{\Lambda}=\displaystyle\frac{\left|\tau(z^{*})_{2}^{2}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{11}-2\tau(z^{*})_{1}\tau(z^{*})_{2}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{12}+\tau(z^{*})_{1}^{2}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{22}\right|}{{\big{(}\tau(z^{*})_{1}^{2}+\tau(z^{*})_{2}^{2}\big{)}}^{3/2}}.$
(30)

Since both $L_{D}$ and $L_{\Lambda}$ are one-dimensional, it follows from (29)
and (30) that

$\det(L_{\Lambda}^{-1}(L_{\Lambda}-L_{D}))=L_{\Lambda}^{-1}(L_{\Lambda}-L_{D})=1-\frac{L_{D}}{L_{\Lambda}}=\kappa(z)^{2}.$

for $\kappa(z)^{2}$ as in (12). It now follows with (28) that

$\displaystyle\displaystyle\int_{D_{z}\cap\,B_{z}}h^{(n)}(x)\,\textup{d}x$
$\displaystyle=\displaystyle\frac{1}{\sqrt{2\pi
n}\,\xi(z)\,\kappa(z)}e^{-n\Lambda_{p}^{*}(z^{*})}\,(1+o(1)).$ (31)

Comparing (31) with the upper bound of the integral outside of $B_{z}$ in
(22), we can see that the integral over $B_{z}^{c}$ is negligible for large
$n\in\mathbb{N}$. Thus, combining (21), (22) and (31) finishes the proof of
Theorem 3.1.∎

##  7\. Proof of the Main Result for $\ell_{p}^{n}$-balls

We use the notation and definitions established in Sections 2 through 4. Let
$1\leq q<p<\infty$ and $z>m_{p,q}$ be such that $z^{*}\in\mathcal{J}_{p}$. We
proceed similarly to the previous proof, using the reformulation of
$\mathbb{P}\left(n^{1/p-1/q}\,\|\mathscr{Z}^{(n)}\|_{q}>z\right)$ from (20) in
conjunction with the density approximation from Proposition 5.2. The resulting
integral over $\mathscr{D}_{z}$ is again split into a neighbourhood of the
minimum of $\mathcal{I}_{\mathscr{S}}$ over $\overline{\mathscr{D}}_{z}$ and
its complement, which, according to Lemma 4.1, is attained at
$z^{**}=(z^{q},1,1)$. For the integral within that neighbourhood, we apply a
result of Breitung and Hohenbichler [12], which yields an integral
approximation under less restrictive differentiability conditions than those
in Proposition 6.1. This result is again geometric in nature, as the behaviour
of the density on $\partial\mathscr{D}_{z}$ still heavily dictates the value
of the overall approximation. However, since this result is formulated for a
certain neighbourhood of the origin, we first need to construct a sufficient
transformation, mapping our deviation area into such a neighbourhood. After
that, we calculate the specific approximation in our setting.

###### Proof of Theorem 3.2.

We assume the set-up of Theorem 3.2 and use the reformulation (20) to proceed
by considering $\mathbb{P}\left(\mathscr{S}^{(n)}\in\mathscr{D}_{z}\right)$.
Let $\mathscr{B}_{z}\subset\mathbb{R}^{3}$ be an open neighbourhood around
$z^{**}=(z^{q},1,1)$ small enough that the first two coordinates of points
within $\mathscr{B}_{z}$ lie in $\mathcal{J}_{p}$ and the third is positive.
Then it holds by Proposition 5.2 that

$\displaystyle\mathbb{P}\left(\mathscr{S}^{(n)}\in\mathscr{D}_{z}\right)$
$\displaystyle=$
$\displaystyle\int_{\mathscr{D}_{z}\cap\,\mathscr{B}_{z}}\mathscr{h}^{(n)}(x_{1},x_{2},y)\,\textup{d}x_{1}\textup{d}x_{2}\,\textup{d}y+\int_{\mathscr{D}_{z}\cap\,\mathscr{B}_{z}^{c}}\mathscr{h}^{(n)}(x_{1},x_{2},y)\,\textup{d}x_{1}\textup{d}x_{2}\,\textup{d}y.\quad$
(32)

As in the proof of Theorem 3.1, we can follow from Lemma 4.1 ii) and the LDP
in Proposition 2.6 that there is an $\eta>0$, such that

$\mathbb{P}\left(\mathscr{S}^{(n)}\in\mathscr{D}_{z}\cap\,\mathscr{B}_{z}^{c}\right)\leq
e^{-n\,\mathcal{I}_{\mathscr{S}}(z^{**})-n\eta}(1+o(1))=\frac{1}{e^{n\,\eta}}\,e^{-n\,\Lambda_{p}^{*}(z^{*})}(1+o(1)),$
(33)

with
$\mathcal{I}_{\mathscr{S}}(t)=\big{[}\Lambda_{p}^{*}(t_{1},t_{2})-\log(t_{3})\big{]},$
as defined in Lemma 2.7. Let us now consider the first integral in (32). Since
$z^{*}\in\mathcal{J}_{p}$, for sufficiently small $\mathscr{B}_{z}$, we have
that $x=(x_{1},x_{2})\in\mathcal{J}_{p}$ and $y\in(0,1]$. By the density
approximation from Proposition 5.2, it then holds that

$\int_{\mathscr{D}_{z}\cap\,B_{z}}\mathscr{h}^{(n)}(x_{1},x_{2},y)\,\textup{d}x_{1}\,\textup{d}x_{2}\,\textup{d}y=\frac{n^{2}}{2\pi}\,\int_{\mathscr{D}_{z}\cap\,B_{z}}y^{-1}{(\det\mathfrak{H}_{x})}^{-1/2}\,e^{-n\,\mathcal{I}_{\mathscr{S}}(x_{1},x_{2},y)}\,\textup{d}x_{1}\,\textup{d}x_{2}\,\textup{d}y\,(1+o(1)).$

As we have seen in Lemma 4.1, $\mathcal{I}_{\mathscr{S}}$ attains its infimum
on $\overline{\mathscr{D}}_{z}$ at $z^{**}$. However, we cannot use the result
of Adriani and Baldi from Proposition 6.1 here, since at $z^{**}$ the boundary
of $\mathscr{D}_{z}\cap\mathscr{B}_{z}$ is not differentiable, and thereby not
smooth. Hence, we use the following asymptotic integral approximation results
based on Breitung and Hohenbichler [12], which gives a Laplace integral
approximation very similar to that in Liao and Ramanan [37, Lemma 5.1], but
under weaker conditions.

######  Proposition 7.1.

Let $F\subset\mathbb{R}^{3}$ be a bounded closed set containing the origin in
its interior. If

  1. (a)

$f:F\to\mathbb{R}$ and $g:F\to\mathbb{R}$ are continuous functions with
$g(\mathbf{0})\neq 0$, where $\mathbf{0}:=(0,0,0),$

  2. (b)

$f(x)>f(\mathbf{0})$ for all $x\in
F\cap(\mathbb{R}^{2}_{+}\times\mathbb{R})\setminus\\{0\\}$,

  3. (c)

there is a neighbourhood $V\subset F$ of $\mathbf{0}$ in which $f$ is twice
continuously differentiable,

  4. (d)

$f_{[1,0,0]}>0$, $f_{[0,1,0]}>0$, and $f_{[0,0,2]}>0$, with derivatives
$f_{[i,j,k]}=f_{[i,j,k]}(\mathbf{0})$ as in (1),

then it holds that

$\int_{F\cap(\mathbb{R}^{2}_{+}\times\mathbb{R})}g(x)e^{-nf(x)}\,\textup{d}x=\frac{\sqrt{2\pi}}{n^{5/2}}\,\frac{g(x^{*})}{f_{[1,0,0]}f_{[0,1,0]}\sqrt{f_{[0,0,2]}}}e^{-nf(x^{*})}(1+o(1)),$

######  Remark 7.2.

This is the result from [12, Lemma 4] for $n=3,k=2$ and functions $g$ and
$(-f)$ instead of $h$ and $f$. The parameter $\lambda$ in our setting is
replaced by the integer $n\in\mathbb{N}$. Furthermore, a typo within said
result has been corrected, namely the sum in [12, Equation (11)] is replaced
by a product (compare proof therein). This proposition is quite close to [37,
Lemma 5.1], but does not require the same level of smoothness of $f$ and $g$,
and $g$ does not depend on $n\in\mathbb{N}$.

To apply this, we use a transformation of
$\mathscr{D}_{z}\cap\mathscr{B}_{z}$, mapping $z^{**}=(z^{q},1,1)$ to
$\mathbf{0}$. Consider

$\mathfrak{I}:\mathbb{R}^{3}\to\mathbb{R}^{3}\qquad\text{ with
}\qquad\mathfrak{I}(x_{1},x_{2},y)=(y^{q}x_{1}-z^{q}x_{2}^{q/p},1-y,x_{2}-1)=(t_{1},t_{2},t_{3}).$

It then holds that $\mathfrak{I}(z^{**})=\mathbf{0}$ and
$\mathfrak{I}(\mathscr{D}_{z})=\tilde{\mathscr{D}}_{z}:=\\{t\in\mathbb{R}^{3}:t_{1}>0,t_{2}\in[0,1),t_{3}>-1\\}$.
Furthermore, in a neighbourhood of $z^{**}$ small enough such that $t_{2}<1$,
$\mathfrak{I}$ is invertible with

$\mathfrak{I}^{-1}(t_{1},t_{2},t_{3})=\left(\frac{t_{1}+z^{q}(t_{3}+1)^{q/p}}{(1-t_{2})^{q}},\,t_{3}+1,\,1-t_{2}\right).$

Let us calculate the Jacobian of $\mathfrak{I}^{-1}$:

$J_{t}\mathfrak{I}^{-1}(t)=\left(\begin{array}[]{ccc}\frac{1}{(1-t_{2})^{q}}&\frac{q(t_{1}+z^{q}(t_{3}+1)^{q/p})}{(1-t_{2})^{q+1}}&\frac{z^{q}\frac{q}{p}(t_{3}+1)^{(q/p)-1}}{(1-t_{2})^{q}}\\\
0&0&1\\\ 0&-1&0\end{array}\right).$ (34)

Thus, we have that $\displaystyle|\det
J_{t}\mathfrak{I}^{-1}(t)|=(1-t_{2})^{-q}$. We set
$\mathscr{g}(x_{1},x_{2},y)=y^{-1}{(\det\mathfrak{H}_{x})}^{-1/2}$, as well as
$\tilde{\mathscr{B}}_{z}:=\mathfrak{I}(\mathscr{B}_{z})$, and transform the
area of integration via $\mathfrak{I}^{-1}$, yielding

$\displaystyle\mathbb{P}\left(\mathscr{S}^{(n)}\in\mathscr{D}_{z}\cap\mathscr{B}_{z}\right)$
$\displaystyle=$
$\displaystyle\int_{\mathscr{D}_{z}\cap\,\mathscr{B}_{z}}\mathscr{h}^{(n)}(x_{1},x_{2},y)\,\textup{d}x_{1}\textup{d}x_{2}\,\textup{d}y$
$\displaystyle=$
$\displaystyle\frac{n^{2}}{2\pi}\,\int_{\mathscr{D}_{z}\cap\,\mathscr{B}_{z}}y^{-1}{(\det\mathfrak{H}_{x})}^{-1/2}\,e^{-n\,[\Lambda_{p}^{*}(x_{1},x_{2})-\log(y)]}\,\textup{d}x_{1}\,\textup{d}x_{2}\,\textup{d}y\,(1+o(1))$
$\displaystyle=$
$\displaystyle\frac{n^{2}}{2\pi}\,\int_{\mathscr{D}_{z}\cap\,\mathscr{B}_{z}}\mathscr{g}(x_{1},x_{2},y)\,e^{-n\,\mathcal{I}_{\mathscr{S}}(x_{1},x_{2},y)}\,\textup{d}x_{1}\,\textup{d}x_{2}\,\textup{d}y\,(1+o(1))$
$\displaystyle=$
$\displaystyle\frac{n^{2}}{2\pi}\,\int_{\tilde{\mathscr{D}}_{z}\cap\,\tilde{\mathscr{B}}_{z}}\mathscr{g}\circ\mathfrak{I}^{-1}(t)\,e^{-n\,\mathcal{I}_{\mathscr{S}}\circ\mathfrak{I}^{-1}(t)}\,(1-t_{2})^{-q}\,\textup{d}t\,(1+o(1)).$

We now set
$\tilde{g}(t):=(1-t_{2})^{-q}\,\mathscr{g}\circ\mathfrak{I}^{-1}(t)$ and
$\tilde{f}(t):=\mathcal{I}_{\mathscr{S}}\circ\mathfrak{I}^{-1}(t)$, then

$\mathbb{P}\left(\mathscr{S}^{(n)}\in\mathscr{D}_{z}\cap\mathscr{B}_{z}\right)=\frac{n^{2}}{2\pi}\,\int_{\tilde{\mathscr{D}}_{z}\cap\,\tilde{\mathscr{B}}_{z}}\tilde{g}(t)\,e^{-n\,\tilde{f}(t)}\,\textup{d}t\,(1+o(1)).$
(35)

We intend to apply Proposition 7.1 to the integral in (35) for
$F=\tilde{\mathscr{B}}_{z}$. It holds that
$\tilde{\mathscr{D}}_{z}\cap\,\tilde{\mathscr{B}}_{z}$ is bounded and since
the value of the integral is the same if we integrate over the open set
$\tilde{\mathscr{B}}_{z}$ or its closure, we will continue to work with
$\tilde{\mathscr{B}}_{z}$. Further, we have that $\tilde{\mathscr{B}}_{z}$
contains the origin in its interior, as the interior point $z^{**}$ of
${\mathscr{B}}_{z}$ is again mapped by the continuous function $\mathfrak{I}$
onto an interior point, which is $\mathfrak{I}(z^{**})=\mathbf{0}$. Since we
have chosen the neighbourhood ${\mathscr{B}}_{z}$ of $z^{**}$ small enough for
$\tilde{\mathscr{B}}_{z}$ to not contain $(t_{1},1,t_{3})$, it holds that

$\tilde{g}(t)=(1-t_{2})^{-q}\,\left[(1-t_{2})^{-1}\,{\left(\det\mathfrak{H}_{(\mathfrak{I}^{-1}(t)_{1},\mathfrak{I}^{-1}(t)_{3})}\right)}^{-1/2}\right]$

is also differentiable on $\tilde{\mathscr{D}_{z}}\cap\tilde{\mathscr{B}}_{z}$
as a composition of differentiable functions and thereby continuous on
$\tilde{\mathscr{B}}_{z}$. The differentiability of $\mathfrak{I}^{-1}$,
together with that of $\Lambda_{p}^{*}$ shown in the proof of Theorem 3.1,
yields the differentiability (and thereby the continuity) of
$\tilde{f}(t):=\mathcal{I}_{\mathscr{S}}\circ\mathfrak{I}^{-1}(t)$ on
$\tilde{\mathscr{B}}_{z}$. It holds furthermore that

$\displaystyle\tilde{g}(\mathbf{0})={\left(\det\mathfrak{H}_{z^{*}}\right)}^{-1/2},$
(36)

which is positive, since $\mathfrak{H}_{z^{*}}$ is positive definite on
$\mathcal{J}_{p}$, as discussed in Section 2. Again, for ${\mathscr{B}}_{z}$
small enough, it also holds (up to a null set) that
$\tilde{\mathscr{B}}_{z}\cap(\mathbb{R}^{2}_{+}\times\mathbb{R})=\tilde{\mathscr{B}}_{z}\cap\tilde{\mathscr{D}}_{z}$,
on which we know from Lemma 2.7 and Lemma 4.1 that
$\mathbf{0}=\mathfrak{I}(z^{**})$ is the unique infimum of $\tilde{f}$ since

$\displaystyle\tilde{f}(\mathbf{0})=\mathcal{I}_{\mathscr{S}}\circ\mathfrak{I}^{-1}(\mathbf{0})=\mathcal{I}_{\mathscr{S}}(z^{**})=\Lambda_{p}^{*}(z^{*}).$
(37)

We can see from (34) that all partial derivatives of $\mathfrak{I}^{-1}$ are
all themselves continuously differentiable in a sufficiently small
neighbourhood of $\mathbf{0}$. Thereby, $\mathfrak{I}^{-1}$ is twice
continuously differentiable in such a neighbourhood. The two-fold continuous
differentiability of $\Lambda_{p}^{*}$ has already been shown in the proof of
Theorem 3.1. Finally, by Lemma 6.2 i), it holds that

$\displaystyle\nabla_{(x_{1},x_{2},y)}\mathcal{I}_{\mathscr{S}}(z^{**})$
$\displaystyle=$ $\displaystyle\left(\frac{\partial}{\partial
x_{1}}\Lambda_{p}^{*}(x_{1},x_{2}),\frac{\partial}{\partial
x_{2}}\Lambda_{p}^{*}(x_{1},x_{2}),-\frac{1}{y}\right)\Big{|}_{(x_{1},x_{2},y)=z^{**}}$
$\displaystyle=$
$\displaystyle\left(\tau(x)_{1},\tau(x)_{2},-\frac{1}{y}\right)\Big{|}_{(x_{1},x_{2},y)=z^{**}}$
$\displaystyle=$
$\displaystyle\left(\tau(z^{*})_{1},\tau(z^{*})_{2},-1\right),$

from which we can deduce that

$\displaystyle\nabla_{t}\tilde{f}(\mathbf{0})\,$ $\displaystyle=$
$\displaystyle\,\nabla_{(x_{1},x_{2},y)}\mathcal{I}_{\mathscr{S}}(z^{**})\,J_{t}\mathfrak{I}^{-1}(\mathbf{0})$
(41) $\displaystyle=$
$\displaystyle\left(\tau(z^{*})_{1},\tau(z^{*})_{2},-1\right)\,\left(\begin{array}[]{ccc}1&qz^{q}&z^{q}\,\frac{q}{p}\\\
0&0&1\\\ 0&-1&0\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\tau(z^{*})_{1},\,\,qz^{q}\tau(z^{*})_{1}+1,\,\,z^{q}\,\frac{q}{p}\tau(z^{*})_{1}+\tau(z^{*})_{2}\right).$
(42)

It thereby follows that $\nabla_{t}\tilde{f}(\mathbf{0})\neq\mathbf{0}$, as
the first two components cannot be equal to zero simultaneously. But since
$\tilde{f}(t)$ attains its infimum on
$\tilde{\mathscr{B}}_{z}\cap(\mathbb{R}^{2}_{+}\times\mathbb{R})$ in
$t=\mathbf{0}$, it holds that $\tilde{f}_{[1,0,0]}>0$ and
$\tilde{f}_{[0,1,0]}>0$, as otherwise a step into either direction
$t_{1},t_{2}$ would maintain or decrease the value of $\tilde{f}$,
contradicting the unique infimum property of $\mathbf{0}$. On the other hand,
by the same argument, it has to hold that $\tilde{f}_{[0,0,1]}=0$ and
$\tilde{f}_{[0,0,2]}>0$, as otherwise a step into either direction
$t_{3},(-t_{3})$ would maintain or decrease $\tilde{f}$, again contradicting
the unique infimum property of $\mathbf{0}$. Hence, we have shown all
conditions for Proposition 7.1, whereby it now follows for the integral in
(35) that

$\displaystyle\mathbb{P}\left(\mathscr{S}^{(n)}\in\mathscr{D}_{z}\cap\mathscr{B}_{z}\right)$
$\displaystyle=$
$\displaystyle\frac{n^{2}}{2\pi}\,\int_{\tilde{\mathscr{D}}_{z}\cap\,\tilde{\mathscr{B}}_{z}}\tilde{g}(t)\,e^{-n\,\tilde{f}(t)}\,\textup{d}t\,(1+o(1))$
(43) $\displaystyle=$
$\displaystyle\frac{n^{2}}{2\pi}\,\int_{\tilde{\mathscr{B}}_{z}\cap(\mathbb{R}^{2}_{+}\times\mathbb{R})}\tilde{g}(t)\,e^{-n\,\tilde{f}(t)}\,\textup{d}t\,(1+o(1))$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2\pi
n}}\,\frac{\tilde{g}(\mathbf{0})}{\tilde{f}_{[1,0,0]}\tilde{f}_{[0,1,0]}\sqrt{\tilde{f}_{[0,0,2]}}}e^{-n\tilde{f}(\mathbf{0})}(1+o(1)).$

The final term that remains to be calculated explicitly is
$\tilde{f}_{[0,0,2]}$, as $\tilde{f}_{[1,0,0]}$ and $\tilde{f}_{[0,1,0]}$ are
given in (41).We start by noting that

$\displaystyle\frac{\partial}{\partial
t_{3}}\mathfrak{I}^{-1}(t)\Big{|}_{t=\mathbf{0}}=\left(\frac{z^{q}\frac{q}{p}(t_{3}+1)^{(q/p)-1}}{(1-t_{2})^{q}},1,0\right)\Big{|}_{t=\mathbf{0}}=\left(z^{q}\frac{p}{q},1,0\right),$

and

$\displaystyle\frac{\partial^{2}}{\partial^{2}t_{3}}\mathfrak{I}^{-1}(t)\big{|}_{t=\mathbf{0}}=\left(\frac{z^{q}\frac{q}{p}\big{(}\frac{q}{p}-1\big{)}(t_{3}+1)^{(q/p)-2}}{(1-t_{2})^{q}},0,0\right)\Big{|}_{t=\mathbf{0}}=\left(\frac{z^{q}q^{2}}{p^{2}}-\frac{z^{q}q}{p},0,0\right).$

By Lemma 6.2 ii), we get that

$\mathcal{H}_{(x_{1},x_{2},y)}\mathcal{I}_{\mathscr{S}}(z^{**})=\left(\begin{array}[]{ccc}\vphantom{\int\limits^{1}}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{11}&\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{12}&0\\\
\vphantom{\int\limits_{0}^{1}}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{21}&\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{22}&0\\\
0&0&y^{-2}\end{array}\right).$

It thereby follows that

$\displaystyle\tilde{f}_{[0,0,2]}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}}{\partial^{2}t_{3}}\mathcal{I}_{\mathscr{S}}\circ\mathfrak{I}^{-1}(\mathbf{0})$
(44) $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
t_{3}}\left[\nabla_{(x_{1},x_{2},y)}\mathcal{I}_{\mathscr{S}}(\mathfrak{I}^{-1}(t))\,\frac{\partial}{\partial
t_{3}}\mathfrak{I}^{-1}(t)\right]\Bigg{|}_{t=\mathbf{0}}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
t_{3}}\left[\nabla_{(x_{1},x_{2},y)}\mathcal{I}_{\mathscr{S}}(\mathfrak{I}^{-1}(t))\right]\Big{|}_{t=\mathbf{0}}\,\,\frac{\partial}{\partial
t_{3}}\mathfrak{I}^{-1}(\mathbf{0})+\nabla_{(x_{1},x_{2},y)}\mathcal{I}_{\mathscr{S}}(z^{**})\,\,\frac{\partial^{2}}{\partial^{2}t_{3}}\mathfrak{I}^{-1}(\mathbf{0})$
$\displaystyle=$
$\displaystyle\left(z^{q}\frac{q}{p},1,0\right)\mathcal{H}_{(x_{1},x_{2},y)}\mathcal{I}_{\mathscr{S}}(z^{**})\,\,\left(z^{q}\frac{q}{p},1,0\right)+\Big{(}\tau(z^{*})_{1},\tau(z^{*})_{2},-1\Big{)}\left(\frac{z^{q}q^{2}}{p^{2}}-\frac{z^{q}q}{p},0,0\right)$
$\displaystyle=$
$\displaystyle\left(\frac{z^{q}q}{p},1,0\right)\left(\frac{z^{q}q}{p}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{11}+\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{12},\frac{z^{q}q}{p}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{21}+\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{22},0\right)+\tau(z^{*})_{1}\Big{(}\frac{z^{q}q^{2}}{p^{2}}-\frac{z^{q}q}{p}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{z^{2q}q^{2}}{p^{2}}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{11}+\frac{2z^{q}q}{p}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{12}+\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{22}+\,\tau(z^{*})_{1}\Big{(}\frac{z^{q}q^{2}}{p^{2}}-\frac{z^{q}q}{p}\Big{)}.\vphantom{\int\limits_{0}}$

Plugging the terms from (36), (41) and (44) into the fraction in (43), we get
that

$\displaystyle\frac{\tilde{g}(\mathbf{0})}{\tilde{f}_{[1,0,0]}\tilde{f}_{[0,1,0]}\sqrt{|\tilde{f}_{[0,0,2]}|}}$
$\displaystyle=$
$\displaystyle{\left(\det\mathfrak{H}_{z^{*}}\right)}^{-1/2}\,(\tau(z^{*})_{1})^{-1}\,(qz^{q}\tau(z^{*})_{1}+1)^{-1}$
(45)
$\displaystyle\times\,{\Bigg{[}\frac{z^{2q}q^{2}}{p^{2}}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{11}+\frac{2z^{q}q}{p}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{12}+\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{22}+\,\tau(z^{*})_{1}\Big{(}\frac{z^{q}q^{2}}{p^{2}}-\frac{z^{q}q}{p}\Big{)}\Bigg{]}}^{-1/2}\vphantom{\int\limits_{0}}$
$\displaystyle=$
$\displaystyle\Bigg{[}\det\mathfrak{H}_{z^{*}}\,\left(\tau(z^{*})_{1}\right)^{2}\,\left(qz^{q}\tau(z^{*})_{1}+1\right)^{2}$
$\displaystyle\times\,\Bigg{(}\frac{z^{2q}q^{2}}{p^{2}}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{11}+\frac{2z^{q}q}{p}\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{12}+\left(\mathfrak{H}_{z^{*}}^{-1}\right)_{22}+\,\tau(z^{*})_{1}\frac{z^{q}q(q-p)}{p^{2}}\Bigg{)}\Bigg{]}^{-1/2}\vphantom{\int\limits_{0}}$
$\displaystyle=$ $\displaystyle\gamma(z)^{-1},$

with $\gamma(z)$ as in (13). Hence, it follows with (37), (43), and (45) that

$\mathbb{P}\left(\mathscr{S}^{(n)}\in\mathscr{D}_{z}\cap\mathscr{B}_{z}\right)=\frac{1}{\sqrt{2\pi
n}\,\gamma(z)}e^{-n\,\Lambda_{p}^{*}(z^{*})}\ (1+o(1)).$ (46)

Combining the representation from (32) with the two integral estimates from
(33) and (46) shows that the integral in the complement of $\mathscr{B}_{z}$
can be neglected and we have that

$\mathbb{P}\left(n^{1/p-1/q}\|\mathscr{Z}^{(n)}\|_{q}>z\right)=\mathbb{P}\left(\mathscr{S}^{(n)}\in\mathscr{D}_{z}\cap\mathscr{B}_{z}\right)=\frac{1}{\sqrt{2\pi
n}\,\gamma(z)}\,e^{-n\,\Lambda_{p}^{*}(z^{*})}\,(1+o(1)),$

which proves our second main result for $\ell_{p}^{n}$-balls. ∎

## Appendix

###### Proof of Lemma 2.5.

Let $z>m_{p,q}$ such that $z^{*}=(z^{q},1)\in\mathcal{J}_{p}$. Then it holds
that

$\displaystyle\mathcal{I}_{\|Z\|}(z)=\inf_{{\text{$t_{1},t_{2}>0$}}\atop{\text{$t_{1}^{1/q}t_{2}^{-1/p}=z$}}}\Lambda_{p}^{*}(t_{1},t_{2})=\inf_{\tilde{t}_{1},\tilde{t}_{2}>0:\\\
\,\tilde{t}_{1}=z\,\tilde{t}_{2}}\Lambda_{p}^{*}(\tilde{t}_{1}^{q},\tilde{t}_{2}^{p})=\inf_{\tilde{t}_{2}>0}\Lambda_{p}^{*}(z^{q}{\tilde{t}_{2}}^{q},{\tilde{t}_{2}}^{p}).$

We set $t_{z}:=(z^{q}\tilde{t}_{2}^{q},\tilde{t}_{2}^{p})$, then with (9) it
follows that

$\displaystyle\mathcal{I}_{\|Z\|}(z)=\inf_{\tilde{t}_{2}>0}\sup_{s\in\mathbb{R}^{2}}\left(\langle
s,t_{z}\rangle-\Lambda_{p}(s)\right)=\inf_{\tilde{t}_{2}>0}\Big{[}\langle\tau(t_{z}),t_{z}\rangle-\Lambda_{p}(\tau(t_{z}))\Big{]}.$

Our goal is to show that the infimum is attained at $t_{z}^{*}:=z^{*}$, i.e.
at $\tilde{t}_{2}=1$. Recall the definition $g_{t}(s):=\langle
s,t\rangle-\Lambda_{p}(s)$ for $t\in\mathcal{J}_{p}$ from Section 2.4. By the
definition of $\tau(t_{z})$ it holds that $g_{t_{z}}(s)$ attains its supremum
at $\tau(t_{z})$, thus it holds that
$\nabla_{s}\,g_{t_{z}}(s)\big{|}_{s=\tau(t_{z})}=t_{z}-\nabla_{s}\Lambda_{p}(s)\big{|}_{s=\tau(t_{z})}=0,$
which gives

$\displaystyle
t_{z}=\left(z^{q}\tilde{t}_{2}^{q},\tilde{t}_{2}^{p}\right)=\left(\frac{\partial}{\partial
s_{1}}\Lambda_{p}(s)\big{|}_{s=\tau(t_{z})},\frac{\partial}{\partial
s_{2}}\Lambda_{p}(s)\big{|}_{s=\tau(t_{z})}\right).$ (47)

We now aim to write $\frac{\partial}{\partial s_{2}}\Lambda_{p}(s)$ with
respect to $\frac{\partial}{\partial s_{1}}\Lambda_{p}(s)$ and then use the
above equations. To do so, we firstly want to reformulate $\Lambda_{p}$ along
the lines of [19, Lemma 5.7]. It holds that

$\displaystyle\Lambda_{p}(s):=\log\int_{\mathbb{R}}e^{s_{1}|y|^{q}+s_{2}|y|^{p}}f_{p}(y)\,\textup{d}y=\log\left(\frac{1}{2p^{1/p}\Gamma\big{(}1+\frac{1}{p}\big{)}}\,\int_{\mathbb{R}}e^{s_{1}|y|^{q}-\frac{1}{p}(1-ps_{2})|y|^{p}}\,\textup{d}y\right).$

The change of variables $x=(1-ps_{2})^{1/p}y$ then gives

$\hskip
2.84544pt\displaystyle\Lambda_{p}(s)=\log\left((1-ps_{2})^{-1/p}\int_{\mathbb{R}}e^{\frac{s_{1}}{(1-ps_{2})^{q/p}}|x|^{q}}f_{p}(x)\,\textup{d}x\right)=-\frac{1}{p}\log(1-ps_{2})+\log\varphi_{|X|^{q}}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right),$

where $\varphi_{|X|^{q}}$ is the m.g.f. of a random variable $|X|^{q}$ with
$X\sim\mathbf{N}_{p}$. Hence,

$\displaystyle\frac{\partial}{\partial s_{1}}\Lambda_{p}(s)$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
s_{1}}\left[\log\varphi_{|X|^{q}}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)\right]$
$\displaystyle=$
$\displaystyle\varphi_{|X|^{q}}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)^{-1}\frac{\partial}{\partial
s_{1}}\left[\varphi_{|X|^{q}}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)\right]$
$\displaystyle=$
$\displaystyle\varphi_{|X|^{q}}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)^{-1}\int_{\mathbb{R}}(1-ps_{2})^{-q/p}\,|x|^{q}\,e^{\frac{s_{1}}{(1-ps_{2})^{q/p}}|x|^{q}}f_{p}(x)\,\textup{d}x$
$\displaystyle=$
$\displaystyle(1-ps_{2})^{-q/p}\varphi_{|X|^{q}}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)^{-1}\varphi_{|X|^{q}}^{\prime}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right),$

where
$\varphi_{|X|^{q}}^{\prime}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)=\varphi_{|X|^{q}}^{\prime}\left(t\right)\Big{|}_{t=\frac{s_{1}}{(1-ps_{2})^{q/p}}}$.
Moreover, with the above we get that

$\displaystyle\frac{\partial}{\partial s_{2}}\Lambda_{p}(s)$ $\displaystyle=$
$\displaystyle{(1-ps_{2})}^{-1}+\frac{\partial}{\partial
s_{2}}\left[\log\varphi_{|X|^{q}}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)\right]$
(48) $\displaystyle=$
$\displaystyle{(1-ps_{2})}^{-1}+\varphi_{|X|^{q}}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)^{-1}\frac{\partial}{\partial
s_{2}}\left[\varphi_{|X|^{q}}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)\right]$
$\displaystyle=$
$\displaystyle{(1-ps_{2})}^{-1}+\varphi_{|X|^{q}}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)^{-1}\int_{\mathbb{R}}\frac{qs_{1}}{(1-ps_{2})^{(q+p)/p}}\,|x|^{q}\,e^{\frac{s_{1}}{(1-ps_{2})^{q/p}}|x|^{q}}f_{p}(x)\,\textup{d}x$
$\displaystyle=$
$\displaystyle{(1-ps_{2})}^{-1}+\frac{qs_{1}}{(1-ps_{2})^{(q+p)/p}}\,\varphi_{|X|^{q}}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)^{-1}\varphi_{|X|^{q}}^{\prime}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)$
$\displaystyle=$
$\displaystyle{(1-ps_{2})}^{-1}+\frac{qs_{1}}{1-ps_{2}}\,(1-ps_{2})^{-q/p}\,\varphi_{|X|^{q}}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)^{-1}\varphi_{|X|^{q}}^{\prime}\left(\frac{s_{1}}{(1-ps_{2})^{q/p}}\right)$
$\displaystyle=$
$\displaystyle{(1-ps_{2})}^{-1}+qs_{1}{(1-ps_{2})}^{-1}\,\frac{\partial}{\partial
s_{1}}\Lambda_{p}(s).$

Plugging in the identities from (47) into (48) it follows for
$(s_{1},s_{2})=\big{(}\tau(t_{z})_{1},\tau(t_{z})_{2}\big{)}$:

$\displaystyle\tilde{t}_{2}^{p}=\displaystyle{(1-p\tau(t_{z})_{2})}^{-1}+q\tau(t_{z})_{1}{(1-p\tau(t_{z})_{2})}^{-1}\,z^{q}\tilde{t}_{2}^{q}.$
(49)

Using this, we can calculate the derivative of $\Lambda_{p}^{*}(t_{z})$ in $t$
(we write $t$ instead of $\tilde{t}_{2}$ for notational brevity), where
$\tau(t_{z})$ is considered as a function in $t$ as well. It holds that

$\displaystyle\frac{\partial}{\partial t}\Lambda_{p}^{*}(t_{z})$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial
t}\Lambda_{p}^{*}(z^{q}t^{q},t^{p})\vphantom{\sum\limits_{0}}$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial t}\big{[}\langle
t_{z},\tau(t_{z})\rangle-\Lambda_{p}(\tau(t_{z}))\big{]}\vphantom{\sum\limits_{0}}$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial
t}\big{[}z^{q}t^{q}\tau(t_{z})_{1}+t^{p}\tau(t_{z})_{2}-\Lambda_{p}(\tau(t_{z}))\big{]}\vphantom{\sum\limits_{0}}$
$\displaystyle=$ $\displaystyle
z^{q}qt^{q-1}\tau(t_{z})_{1}+z^{q}t^{q}\frac{\partial}{\partial
t}\tau(t_{z})_{1}+pt^{p-1}\tau(t_{z})_{2}+t^{p}\frac{\partial}{\partial
t}\tau(t_{z})_{2}-\frac{\partial}{\partial
t}\Lambda_{p}(\tau(t_{z}))\vphantom{\sum\limits_{0}}$ $\displaystyle=$
$\displaystyle z^{q}qt^{q-1}\tau(t_{z})_{1}+z^{q}t^{q}\frac{\partial}{\partial
t}\tau(t_{z})_{1}+pt^{p-1}\tau(t_{z})_{2}+t^{p}\frac{\partial}{\partial
t}\tau(t_{z})_{2}-J_{t}(\tau(t_{z}))\nabla_{s}\Lambda_{p}(s)\big{|}_{s=\tau(t_{z})}\vphantom{\sum\limits_{0}}$
$\displaystyle=$ $\displaystyle
z^{q}qt^{q-1}\tau(t_{z})_{1}+z^{q}t^{q}\frac{\partial}{\partial
t}\tau(t_{z})_{1}+pt^{p-1}\tau(t_{z})_{2}+t^{p}\frac{\partial}{\partial
t}\tau(t_{z})_{2}\vphantom{\sum\limits_{0}}$
$\displaystyle-\frac{\partial}{\partial
t}\tau(t_{z})_{1}\frac{\partial}{\partial
s_{1}}\Lambda_{p}(s)\big{|}_{s=\tau(t_{z})}-\frac{\partial}{\partial
t}\tau(t_{z})_{2}\frac{\partial}{\partial
s_{2}}\Lambda_{p}(s)\big{|}_{s=\tau(t_{z})}.$

We now use the identity from (47), which yields

$\displaystyle\frac{\partial}{\partial t}\Lambda_{p}^{*}(t_{z})$
$\displaystyle=$ $\displaystyle
z^{q}qt^{q-1}\tau(t_{z})_{1}+z^{q}t^{q}\frac{\partial}{\partial
t}\tau(t_{z})_{1}+pt^{p-1}\tau(t_{z})_{2}+t^{p}\frac{\partial}{\partial
t}\tau(t_{z})_{2}-\frac{\partial}{\partial
t}\tau(t_{z})_{1}z^{q}t^{q}-\frac{\partial}{\partial
t}\tau(t_{z})_{2}t^{p}\vphantom{\sum\limits_{0}}$ (50) $\displaystyle=$
$\displaystyle z^{q}qt^{q-1}\tau(t_{z})_{1}+pt^{p-1}\tau(t_{z})_{2}.$

Reformulating the identity in (49) yields

$\displaystyle
t^{p}=\displaystyle{(1-p\tau(t_{z})_{2})}^{-1}+q\tau(t_{z})_{1}{(1-p\tau(t_{z})_{2})}^{-1}\,z^{q}t^{q}\Leftrightarrow(1-p\tau(t_{z})_{2})t^{p-1}-t^{-1}=z^{q}t^{q-1}q\tau(t_{z})_{1}.$
(51)

Thus, if we set $\frac{\partial}{\partial t}\Lambda_{p}^{*}(t_{a})=0$, we get
from (50) and (51) that

$\displaystyle\frac{\partial}{\partial t}\Lambda_{p}^{*}(t_{z})=0$
$\displaystyle\Leftrightarrow$ $\displaystyle
0=z^{q}qt^{q-1}\tau(t_{z})_{1}+pt^{p-1}\tau(t_{z})_{2}\vphantom{\sum}$
$\displaystyle\Leftrightarrow$ $\displaystyle
0=(1-p\tau(t_{z})_{2})t^{p-1}-t^{-1}+pt^{p-1}\tau(t_{z})_{2}\vphantom{\sum\limits_{0}}$
$\displaystyle\Leftrightarrow$ $\displaystyle t=1.$

Hence, the infimum of $\Lambda_{p}^{*}$ over $\partial D_{z}$ is attained at
$t_{z}^{*}=(z^{q},1)=z^{*}$ Since $\Lambda_{p}^{*}$ is strictly convex (see
properties of the Legendre-Fenchel transform), this minimum is unique.
Thereby, our claim is proven. ∎

###### Proof of Lemma 2.7.

Let $z>m_{p,q}$ such that $z^{*}=(z^{q},1)\in\mathcal{J}_{p}$. Furthermore,
set $z^{**}:=(z^{q},1,1)$ and
$\mathcal{I}_{\mathscr{S}}(t):=[\Lambda_{p}^{*}(t_{1},t_{2})-\log(t_{3})]$,
$t\in\mathbb{R}^{3}$. We use the definitions of $\mathcal{I}_{\|Z\|}$ and
$\mathcal{I}_{U}$, together with Lemma 2.5, to get that

$\displaystyle\mathcal{I}_{\|\mathscr{Z}\|}(z)$ $\displaystyle=$
$\displaystyle\displaystyle\inf_{{\text{$z=t_{1}^{1/q}t_{2}^{-1/p}t_{3}$}}\atop{\text{$t_{1},t_{2}>0,t_{3}\in(0,1]$}}}\mathcal{I}_{\mathscr{S}}(t)$
$\displaystyle=$
$\displaystyle\displaystyle\inf_{{\text{$z=z_{1}z_{2}$}}\atop{\text{$z_{1}>0,z_{2}\in(0,1]$}}}\left[\inf_{{\text{$t_{1},t_{2}>0$}}\atop{\text{$t_{1}^{1/q}t_{2}^{-1/p}=z_{1}$}}}\Lambda_{p}^{*}(t_{1},t_{2})+\mathcal{I}_{U}(z_{2})\right]$
$\displaystyle=$
$\displaystyle\displaystyle\inf_{{\text{$z=z_{1}z_{2}$}}\atop{\text{$z_{1}>0,z_{2}\in(0,1]$}}}\left[\Lambda_{p}^{*}(z_{1}^{q},1)-\log(z_{2})\right].$

By the same arguments as in the proof of Lemma 4.1, we know that
$\mathcal{I}_{\|Z\|}(z)=\Lambda_{p}^{*}(z^{q},1)$ is strictly convex in $z$ on
$\mathcal{J}_{p}$ with a unique root in $m_{p,q}$. Hence, it follows that for
$z>m_{p,q}$ with $z\in\mathcal{J}_{p}$ it holds that
$\mathcal{I}_{\|Z\|}(z)=\Lambda_{p}^{*}(z^{q},1)$ is strictly increasing in
$z$. Since $z_{2}\leq 1$, $z=z_{1}z_{2}$, and $1<q$, we have $z_{1}^{q}\geq
z>m_{p,q}$, meaning that $\Lambda_{p}^{*}(z_{1}^{q},1)$ is strictly increasing
in $z_{1}$. Furthermore, we can see that $-\log(z_{2})$ is strictly decreasing
in $z_{2}$. Hence, rewriting $z_{1}$ with respect to $z_{2}$ then gives

$\displaystyle\mathcal{I}_{\|\mathscr{Z}\|}(z)$ $\displaystyle=$
$\displaystyle\inf_{{\text{$z_{1}=z/z_{2}$}}\atop{\text{$z_{2}\in(0,1]$}}}\left[\Lambda_{p}^{*}\Big{(}\Big{(}\frac{z}{z_{2}}\Big{)}^{q},1\Big{)}-\log(z_{2})\right],$

which is strictly decreasing in $z_{2}$. Thus, choosing $z_{2}=1$ gives
$z_{1}=z$ and

$\mathcal{I}_{\|\mathscr{Z}\|}(z)=\mathcal{I}_{\mathscr{S}}(z^{**})=\Lambda_{p}^{*}(z^{*}),$

finishing the proof. ∎

## Acknowledgments

The author would like to thank Kavita Ramanan and Joscha Prochno for the
insightful exchanges on the topic of sharp large deviations in asymptotic
geometric analysis. Furthermore, the author would like to thank his supervisor
Christoph Thäle for the helpful discussions, feedback and constructive
criticism throughout the writing of this paper.

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