# A probabilistic approach to Lorentz balls

Zakhar Kabluchko, Joscha Prochno and Mathias Sonnleitner

###### Abstract

We develop a probabilistic approach to study the volumetric and geometric
properties of unit balls $\mathbb{B}_{q,1}^{n}$ of finite-dimensional Lorentz
sequences spaces $\ell_{q,1}^{n}$. More precisely, we show that the empirical
distribution of a random vector $X^{(n)}$ uniformly distributed on the volume
normalized Lorentz ball in $\mathbb{R}^{n}$ converges weakly to a compactly
supported symmetric probability distribution with explicitly given density; as
a consequence we obtain a weak Poincaré-Maxwell-Borel principle for any fixed
number $k\in\mathbb{N}$ of coordinates of $X^{(n)}$ as $n\to\infty$. Moreover,
we prove a central limit theorem for the largest coordinate of $X^{(n)}$,
demonstrating a quite different behavior than in the case of the
$\ell_{q}^{n}$ balls, where a Gumbel distribution appears in the limit. Last
but not least, we prove a Schechtman-Schmuckenschläger type result for the
asymptotic volume of intersections of volume normalized Lorentz and
$\ell^{n}_{p}$ balls. Keywords. Asymptotic volume, central limit theorem,
concentration of measure, convex body, Lorentz space, maximum entropy
principle, Poincaré-Maxwell-Borel principle
MSC. Primary 46B09, 52A23, 60F05; Secondary 46B06, 46B20, 46B45, 94A17

##  1 Introduction

The last decades have revealed a deep connection between the geometry of high-
dimensional convex bodies and probability theory, and each field fruitfully
influenced the other. Probabilistic methods naturally come into play when one
considers a (high-dimensional) convex body in $\mathbb{R}^{n}$, i.e., a
compact and convex set with non-empty interior, as a probability space when it
is endowed with the canonical Borel $\sigma$-field and the normalized uniform
measure.

Probably the most prominent family of convex bodies are the unit balls
$\mathbb{B}_{p}^{n}$ of the classical finite-dimensional sequence spaces
$\ell_{p}^{n}$ ($1\leq p\leq\infty$). This parametric family of bodies is
arguably one the most studied ones in geometric functional analysis and their
analytic and geometric properties are quite well understood today. In numerous
instances it is the already mentioned probabilistic point of view that gives
access to understanding the asymptotic structure as the space dimension $n$
tends to infinity, because there is a rather simple probabilistic
representation of the uniform distribution on $\mathbb{B}_{p}^{n}$. This
representation allows one to go from a random vector uniformly distributed on
$\mathbb{B}_{p}^{n}$, and thus from one having dependent coordinates for
$p<\infty$, to a random vector whose entries are independent and identically
distributed according to the so-called $p$-Gaussian distribution. This heavily
facilitates computations and is therefore one of the key tools used in the
study of $\ell_{p}^{n}$ balls. It were Schechtman and Zinn [56], and
independently Rachev and Rüschendorf [51], who showed that if
$X=(X_{1},\dots,X_{n})$ is distributed uniformly at random on
$\mathbb{B}_{p}^{n}$, then

$\displaystyle X\stackrel{{\scriptstyle{\rm
d}}}{{=}}U^{1/n}\frac{Y}{\|Y\|_{p}},$ (1)

where $U$ is distributed uniformly on $[0,1]$ and $Y=(Y_{1},\dots,Y_{n})$ is
independent of $U$ with $Y_{1},\dots,Y_{n}$ being independent and identically
distributed with Lebesgue density on $\mathbb{R}$ given by

$f_{p}(x):=\begin{cases}\frac{1}{2p^{1/p}\Gamma(1+\frac{1}{p})}e^{-|x|^{p}/p}&:1\leq
p<\infty\cr\frac{1}{2}\mathbbm{1}_{[-1,1]}(x)&:p=\infty.\end{cases}$

Here and below, $\stackrel{{\scriptstyle{\rm d}}}{{=}}$ denotes equality in
distribution. The previous result was lifted in [7] to a wider class of
distributions related to $\ell_{p}^{n}$ balls that include the uniform
distribution and the distribution with respect to the cone probability measure
as special cases. As we mentioned above, the representation presented as well
as its generalization are frequently used in the asymptotic analysis of
geometric and volumetric aspects of those unit balls and we refer, for
instance, to [3, 4, 22, 30, 32, 46, 47, 55, 57] and the survey article [50].

A natural and frequently studied generalization of $\ell_{p}$ spaces (and
their function space counterparts), which is classical not only in functional
analysis [12, 40], harmonic analysis [25] and optimization [38], is the class
of Orlicz spaces. The finite-dimensional Orlicz space $\ell_{M}^{n}$ is
$\mathbb{R}^{n}$ endowed with the norm

$\|(x_{i})_{i=1}^{n}\|_{M}:=\inf\Bigg{\\{}\rho>0\,:\,\sum_{i=1}^{n}M\Big{(}\frac{|x_{i}|}{\rho}\Big{)}\leq
1\Bigg{\\}},$

where $M:\mathbb{R}\to\mathbb{R}$ is symmetric, convex, and satisfies both
$M(0)=0$ and $M(x)>0$ for $x\neq 0$ (when $M(t)=|t|^{p}$, we have
$\|\cdot\|_{M}=\|\cdot\|_{p}$). As in the case of $\ell_{p}^{n}$ spaces, the
Orlicz spaces belong to the important class of finite-dimensional
$1$-symmetric Banach spaces, i.e., spaces $(X,\|\cdot\|_{X})$ with a basis
$e_{1},\dots,e_{n}\in X$ such that

$\Big{\|}\sum_{i=1}^{n}x_{i}e_{i}\Big{\|}_{X}=\Big{\|}\sum_{i=1}^{n}\varepsilon_{i}x_{\pi(i)}e_{i}\Big{\|}_{X}$

holds for all $x_{1},\dots,x_{n}\in\mathbb{R}$, all signs
$\varepsilon_{1},\dots,\varepsilon_{n}\in\\{-1,1\\}$, and all permutations
$\pi$ of the numbers $\\{1,\dots,n\\}$; Orlicz spaces are intensively studied
in the functional analysis literature and we refer to [24, 34, 39, 49, 62] and
references cited therein. Trying to lift or generalize results that can be
obtained for the spaces $\ell_{p}^{n}$ by employing the Schechtman-Zinn
probabilistic representation (1) one soon hits a dead end, because such
representation in the case of unit balls $\mathbb{B}_{M}^{n}$ of Orlicz spaces
$\ell_{M}^{n}$ is not known. As a consequence, generalizations of results
regarding the asymptotic structure of $\mathbb{B}_{p}^{n}$ (as mentioned
above) remained inaccessible for quite a while. Recently, Kabluchko and
Prochno [31], using maximum entropy considerations in the framework of non-
interacting particles that have their origin in statistical mechanics, derived
an _asymptotic_ version of a Schechtman-Zinn type representation for Orlicz
spaces, relating the uniform distribution on Orlicz balls to Gibbs
distributions with potential given by the Orlicz functions. This connection
allowed to obtain a number of further results in the general setting of Orlicz
balls, wich can be found, for instance, in [2, 8, 21, 28, 35].

Alongside Orlicz spaces, the second natural class of generalizations of
$\ell_{p}$ spaces are Lorentz spaces, which were introduced by George Lorentz
in the 1950s [41, 42]. In fact, it had already been observed by Marcinkiewicz
in [43] that Lebesgue spaces are not sufficient to capture the fine properties
of operators on $L_{p}$ spaces. Since their introduction, Lorentz spaces have
found numerous applications in different areas of mathematics such as
approximation theory [16], harmonic analysis [23], interpolation theory [9],
and signal processing [20]. We are interested in the finite-dimensional
Lorentz space $\ell_{q,p}^{n}$ with $1\leq p\leq q\leq\infty$, which is merely
$\mathbb{R}^{n}$ with the norm

$\|(x_{i})_{i=1}^{n}\|_{q,p}:=\Big{(}\sum_{i=1}^{n}|i^{1/q-1/p}x_{i}^{*}|^{p}\Big{)}^{1/p},$

where $x_{1}^{*},\dots,x_{n}^{*}$ is the non-increasing rearrangement of the
numbers $|x_{1}|,\dots,|x_{n}|$ (when $q=p$, we have
$\|\cdot\|_{q,p}=\|\cdot\|_{p}$). Those spaces again belong to the important
class of $1$-symmetric Banach spaces and have also been intensively studied in
the functional analysis literature, for instance, in [5, 18, 26, 27, 48, 53,
54, 60, 61, 64]. Indeed, unexpected phenomena can occur in these spaces; for
example, as proved in [19], they are a counterexample to a conjecture on the
interpolation behaviour of entropy numbers which holds true for $\ell_{p}$
spaces. Results regarding a probabilistic approach to the asymptotic
volumetric and geometric structure of unit balls $\mathbb{B}_{q,p}^{n}$ in
Lorentz spaces $\ell_{q,p}^{n}$, of the same flavor as the ones mentioned for
$\mathbb{B}_{p}^{n}$ or $\mathbb{B}_{M}^{n}$ above, are completely absent from
the literature. One reason is arguably that the non-increasing rearrangement
that appears in the definition of the norm makes any analysis quite delicate.

The motivation of our paper is thus twofold, namely to develop a first
probabilistic approach on the one hand and apply this to study some asymptotic
properties of Lorentz spaces $\ell_{q,p}^{n}$ on the other. In view of the
vast literature regarding a probabilistic take on $\ell_{p}^{n}$ or more
generally $\ell_{M}^{n}$ spaces and their unit balls, this is a first step
towards approaching numerous natural questions that arise, like central limit
theorems, conditional limit theorems, or moderate and large deviations [3, 4,
21, 22, 30, 32, 36], asymptotic volume ratios and thin-shell measure
concentration [2, 31], asymptotic independence of coordinates [8] or
asymptotic thin-shell results [35], just to mention a few.

##  2 Main results

We shall now present our main results, starting with the probabilistic
approach to the uniform distribution on Lorentz balls and continuing with the
applications to geometric and volumetric properties. The results are
restricted to the case of Lorentz balls with parameters $p=1$ and
$1<q\leq\infty$, but we present a conjecture together with heuristic arguments
on what asymptotic probabilistic representation to expect in the general
setting.

We shall denote by

$\widetilde{\mathbb{D}}_{q,p}^{n}:=n^{1/q}\mathbb{B}_{q,p}^{n}=\Bigg{\\{}x\in\mathbb{R}^{n}\colon\frac{1}{n}\sum_{i=1}^{n}\Big{(}\frac{i}{n}\Big{)}^{p/q-1}|x_{i}^{*}|^{p}\leq
1\Bigg{\\}}$

for $q<\infty$ and

$\widetilde{\mathbb{D}}_{\infty,1}^{n}:=\log(n+1)\mathbb{B}_{\infty,1}^{n}=\Bigg{\\{}x\in\mathbb{R}^{n}\colon\frac{1}{\log(n+1)}\sum_{i=1}^{n}\frac{1}{i}|x_{i}^{*}|\leq
1\Bigg{\\}}.$

for $q=\infty$ the normalized unit balls such that, asymptotically up to
constants, they have Lebesgue volume 1. Indeed, it follows from a classical
result on the volume of unit balls in Banach spaces due to Schütt [59] that
$\mathrm{vol}_{n}(\mathbb{B}_{q,1}^{n})^{1/n}\asymp_{q}n^{1/q}$ and, as shown
in [18, Theorem 7], for $q=\infty$ we have
$\mathrm{vol}_{n}(\mathbb{B}_{\infty,1}^{n})^{1/n}\asymp\log(n+1)^{-1}$; here
$\asymp_{q}$ denotes equivalence up to constants depending only on $q$ and
$\asymp$ equivalence up to absolute constants.

The following stochastic representation constitutes the basis for our results
and is of independent interest. It is the probabilistic entry point into the
study of the asymptotic structure of Lorentz balls in the case $p=1$.

######  Theorem A.

Let $1\leq q\leq\infty$, $n\in\mathbb{N}$ and assume $X^{(n)}$ is uniformly
distributed on $\mathbb{B}_{q,1}^{n}$. Then

$X^{(n)}\overset{\rm
d}{=}\Bigg{(}\varepsilon_{1}\frac{\sum_{j=\pi(1)}^{n}\kappa_{q}(j)^{-1}E_{j}}{\sum_{j=1}^{n+1}E_{j}},\dots,\varepsilon_{n}\frac{\sum_{j=\pi(n)}^{n}\kappa_{q}(j)^{-1}E_{j}}{\sum_{j=1}^{n+1}E_{j}}\Bigg{)},$

where $E_{1},\dots,E_{n+1}$ are standard exponential random variables,
$\varepsilon=(\varepsilon_{1},\dots,\varepsilon_{n})$ is uniformly distributed
on $\\{-1,1\\}^{n}$, $\pi$ is uniformly distributed on the symmetric group
$S_{n}$ of permutations of $\\{1,\dots,n\\}$, and for $j\in\\{1,\dots,n\\}$,

$\kappa_{q}(j):=\sum_{i=1}^{j}i^{1/q-1}.$

All random objects are independent of each other.

According to Theorem A, the first (or any) coordinate of a random vector
uniformly distributed on
$\widetilde{\mathbb{D}}_{q,1}^{n}=n^{1/q}\mathbb{B}_{q,1}^{n}$ for $q<\infty$
is equal in distribution to

$\varepsilon_{1}\frac{\frac{1}{n}\sum_{i=u_{n}}^{n}\frac{n^{1/q}}{\kappa_{q}(i)}E_{i}}{\frac{1}{n}\sum_{i=1}^{n+1}E_{i}},$
(2)

where $u_{n}$ is uniformly distributed on $\\{1,\dots,n\\}$ and
$\varepsilon_{1}$ is uniformly distributed on $\\{-1,1\\}$.

######  Remark 1.

Note that the special case $q=1$ of Theorem A corresponds to the
$\ell_{1}^{n}$ ball and is consistent to the well-known fact that the order
statistics of $n$ independent standard exponential random variables are
distributed as

$\frac{E_{n}}{n},\frac{E_{n}}{n}+\frac{E_{n-1}}{n-1},\dots,\sum_{j=1}^{n}\frac{E_{j}}{j}$

beginning with the smallest ($n^{\text{th}}$) order statistic up to the
largest ($1^{\text{st}}$) (see, e.g., [14, Eq. (2.5.5)]).

As a consequence of the proof of Theorem A, we can deduce the following
precise asymptotics for the volume radius; the following can also be obtained
by refining the calculations from [18].

######  Corollary 1.

Let $1\leq q\leq\infty$. As $n\to\infty$, we have

$\mathrm{vol}_{n}(\mathbb{B}_{q,1}^{n})^{1/n}\sim\begin{cases}\frac{2}{q}e^{1/q}n^{-1/q}&:1\leq
q<\infty,\\\ 2(\log n)^{-1}&:q=\infty.\end{cases}$ (3)

In our next theorem, we establish a weak convergence result for the empirical
distribution. It is also used to establish the asymptotic distribution of a
single coordinate presented afterwards.

######  Theorem B.

Let $1<q\leq\infty$ and for each $n\in\mathbb{N}$ assume that
$\widetilde{X}^{(n)}=(\widetilde{X}_{1}^{(n)},\dots,\widetilde{X}_{n}^{(n)})$
is a random vector uniformly distributed on
$\widetilde{\mathbb{D}}_{q,1}^{n}$. Then, for every bounded continuous
function $f\colon\mathbb{R}\to\mathbb{R}$, we have

$\frac{1}{n}\sum_{i=1}^{n}f(\widetilde{X}_{i}^{(n)})\xrightarrow[n\to\infty]{\mathbb{P}}\int_{\mathbb{R}}f(x)\,\nu_{q,1}(\mathrm{d}x),$

where $\nu_{q,1}$ is a symmetric probability measure on $\mathbb{R}$ with
Lebesgue density $f_{q,1}\colon\mathbb{R}\to\mathbb{R}$ given by

$f_{q,1}(x):=\frac{1}{2}\begin{cases}q(1-(q-1)|x|)^{1/(q-1)}\mathbbm{1}_{[-\frac{1}{q-1},\frac{1}{q-1}]}(x)&\colon
q<\infty\\\ \mathbbm{1}_{[-1,1]}(x)&\colon q=\infty.\end{cases}$

Essentially due to exchangeability of the coordinates, it follows from Theorem
B that any fixed choice of $k\in\mathbb{N}$ coordinates is asymptotically
distributed as the $k$-fold product of $\nu_{q,1}$ as the dimension of the
ambient space tends to infinity. We will obtain the following weak Poincaré-
Maxwell-Borel principle for normalized Lorentz balls.

######  Corollary 2.

Let $1<q\leq\infty$ and for each $n\in\mathbb{N}$ assume that
$\widetilde{X}^{(n)}=(\widetilde{X}_{1}^{(n)},\dots,\widetilde{X}_{n}^{(n)})$
is uniformly distributed on $\widetilde{\mathbb{D}}_{q,1}^{n}$. For every
$k\in\mathbb{N}$ and with $\nu_{q,1}$ as in Theorem B, for any bounded and
continuous function $f\colon\mathbb{R}^{k}\to\mathbb{R}$, we have

$\mathbb{E}\big{[}f(\widetilde{X}_{1}^{(n)},\dots,\widetilde{X}_{k}^{(n)})\big{]}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\int_{\mathbb{R}^{k}}f(x)\,\nu_{q,1}^{\otimes
k}(\mathrm{d}x),$

that is, $(\widetilde{X}_{1}^{(n)},\dots,\widetilde{X}_{k}^{(n)})$ converges
in distribution to a vector $Y^{(k)}=(Y_{1},\dots,Y_{k})\sim\nu_{q,1}^{\otimes
k}$.

######  Remark 2.

The case $q=1$ for which $\mathbb{B}_{1}^{n}=\mathbb{B}_{1,1}^{n}$ is already
known [51] and in this case the asymptotic coordinate distributions are two-
sided exponential. By considering $q_{N}:=1+1/N\to 1$, we see that it is
consistent with our result due to limiting behavior

$f_{q_{N}}(x)=\frac{1+1/N}{2}(1-|x|/N)^{N}\mathbbm{1}_{[-N,N]}(x)\stackrel{{\scriptstyle
N\to\infty}}{{\longrightarrow}}\frac{1}{2}\exp(-|x|),\quad x\in\mathbb{R}.$

######  Remark 3.

A notable difference in the asymptotic probabilistic behavior of vectors
chosen uniformly at random from the volume normalized versions of
$\mathbb{B}_{q,1}^{n}$ and $\mathbb{B}_{q}^{n}$ is that in the former case the
asymptotic distribution of a single coordinate has compact support whereas in
the latter case one obtains unbounded support (see (1) and the definition of
the $q$-Gaussian distribution).

In contrast to the maximum norm of uniform random vectors in normalized
$\ell_{q}^{n}$ balls, which has Gumbel fluctuations [32, Theorem 1.1 (c)], we
have a central limit theorem for the Lorentz balls for $q>2$, i.e., while the
volume radius in terms of the dimensions is the same in both cases, the
probabilistic behavior is quite different. In between, that is, for
$q\in(1,2]$, we have a different type of limit theorem which seems to
interpolate between the Gumbel distribution and the normal distribution.

######  Theorem C.

Let $1\leq q<\infty$, $r\in(0,\infty)$ and for each $n\in\mathbb{N}$ assume
that $\widetilde{X}^{(n)}$ is uniformly distributed on
$\widetilde{\mathbb{D}}_{q,1}^{n}$. Then we have:

  1. (i)

For $1\leq q<2$

$n^{1-1/q}\big{(}\|\widetilde{X}^{(n)}\|_{\infty}-\mu_{q,n}\big{)}\xrightarrow[n\to\infty]{{\rm
d}}R_{q}\overset{{\rm d}}{=}\sum_{j=1}^{\infty}\frac{E_{j}-1}{\kappa_{q}(j)},$

  2. (ii)

For $q=2$

$\frac{\sqrt{n}}{\sqrt{\log
n}}\big{(}\|\widetilde{X}^{(n)}\|_{\infty}-\mu_{2,n}\big{)}\xrightarrow[n\to\infty]{{\rm
d}}R_{2}\sim\mathscr{N}(0,1/4),$

  3. (iii)

For $2<q<\infty$

$\sqrt{n}\big{(}\|\widetilde{X}^{(n)}\|_{\infty}-\mu_{q,n}\big{)}\xrightarrow[n\to\infty]{{\rm
d}}\mathscr{N}(0,\sigma_{q}^{2}).$

Here, $R_{1}+\gamma$ has a Gumbel law with distribution function $x\mapsto
e^{-e^{-x}}$, $\gamma$ is the Euler-Mascheroni constant,

$\mu_{q,n}:=\frac{1}{n}\sum_{j=1}^{n}\frac{n^{1/q}}{\kappa_{q}(j)}\qquad\text{and}\qquad\sigma_{q}^{2}:=\frac{1}{q(q-1)^{2}(q-2)}.$

We can also use the probabilistic representation to study the $\ell_{r}^{n}$
norm of a random vector uniformly distributed in
$\mathbb{D}_{q,1}^{n}:=\mathrm{vol}_{n}(\mathbb{B}_{q,1}^{n})^{-1/n}\mathbb{B}_{q,1}^{n}$,
i.e., we determine its asymptotic length with respect to $\|\cdot\|_{r}$; we
shall apply that result later to study the volumetric behavior of certain
intersections. We have the following weak law of large numbers.

######  Theorem D.

Let $1<q\leq\infty$, $1<r\leq\infty$, and for each $n\in\mathbb{N}$ assume
that $X^{(n)}$ is uniformly distributed on $\mathbb{D}_{q,1}^{n}$. Then

$n^{-1/r}\|X^{(n)}\|_{r}\xrightarrow[n\to\infty]{\mathbb{P}}m_{q,r},$

where for $r<\infty$ we have

$m_{q,r}:=\frac{1}{2e^{1/q}}\frac{q}{q-1}\left(\frac{\Gamma(r+1)\Gamma\Big{(}1+\frac{q}{q-1}\Big{)}}{\Gamma\Big{(}r+1+\frac{q}{q-1}\Big{)}}\right)^{1/r}\text{
if }q<\infty\quad\text{and}\quad
m_{\infty,r}:=\frac{1}{2}\Big{(}\frac{1}{r+1}\Big{)}^{1/r}\text{ if
}q=\infty,$

and for $r=\infty$ we have

$m_{q,\infty}:=\frac{1}{2e^{1/q}}\frac{q}{q-1}\text{ if
}q<\infty\quad\text{and}\quad m_{\infty,\infty}:=\frac{1}{2}.$

As a consequence we can deduce the following Schechtman-Schmuckenschläger type
result on the asymptotic volume of intersections of Lorentz and $\ell_{r}^{n}$
balls, thereby complementing the results from [29, 30, 31, 32, 33, 55, 56,
58]. In the following corollary, we shall denote
$\mathbb{D}_{r}^{n}:=\mathrm{vol}_{n}(\mathbb{B}_{r}^{n})^{-1/n}\mathbb{B}_{r}^{n}$.

######  Corollary 3.

Let $1<q\leq\infty$ and $1<r\leq\infty$. Then, for all $t>0$,

$\mathrm{vol}_{n}(\mathbb{D}_{q,1}^{n}\cap
t\mathbb{D}_{r}^{n})\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\begin{cases}1&\colon A_{q,r}t>1\\\ 0&\colon
A_{q,r}t<1,\\\ \end{cases}$

where for $r<\infty$, we have

$A_{q,r}:=\begin{cases}\frac{e^{1/q-1/r}\frac{q-1}{q}}{\Gamma(1+1/r)r^{1/r}}\left(\frac{\Gamma(r+1+\frac{q}{q-1})}{\Gamma(r+1)\Gamma(1+\frac{q}{q-1})}\right)^{1/r}&\colon
q<\infty\\\ \frac{1}{\Gamma(1+1/r)}\left(\frac{r+1}{re}\right)^{1/r}&\colon
q=\infty,\\\ \end{cases}$

and for $r=\infty$ we have

$A_{q,\infty}:=\begin{cases}e^{1/q}\frac{q-1}{q}&\colon q<\infty\\\ 1&\colon
q=\infty.\\\ \end{cases}$

######  Remark 4.

Comparing with analogous results for the $\ell_{p}^{n}$ balls obtained in
[55], we see that for $q=\infty$ and $r<\infty$ the obtained threshold is the
same. Therefore, asymptotically $\mathbb{D}_{\infty,1}^{n}$ behaves somewhat
like $\mathbb{D}_{\infty}^{n}$ when intersected with an $\ell_{r}^{n}$ ball.

Also for $r<\infty$, using $\Gamma(x+\alpha)\sim\Gamma(x)x^{\alpha}$ for
$x\to\infty$, we obtain that

$\lim_{q\to
1}A_{q,r}=\frac{e^{1-1/r}}{\Gamma(1+1/r)\Gamma(r+1)^{1/r}r^{1/r}},$

which equals the threshold one would get for $\mathbb{D}_{1}^{n}$ intersected
with $\mathbb{D}_{r}^{n}$. That is, in the boundary cases $q\in\\{1,\infty\\}$
we have that $\mathbb{D}_{q,1}^{n}$ behaves similarly to $\mathbb{D}_{q}^{n}$.
In between, however, the behaviour of the threshold constants appears to be
different, for example as $q\to\infty$ the constant $A_{q,r}$ grows for
$\mathbb{D}_{q}^{n}$ but converges for $\mathbb{D}_{q,1}^{n}$.

For parameters $p>1$, we currently do not have a probabilistic representation.
However, we can use heuristic arguments based on maximum entropy
considerations in order to derive the following conjecture about the limiting
distribution.

######  Conjecture 1.

Let $1\leq p\leq q<\infty$ and for each $n\in\mathbb{N}$ assume that
$\widetilde{X}^{(n)}=(\widetilde{X}^{(n)}_{1},\dots,\widetilde{X}^{(n)}_{n})$
is uniformly distributed on $\widetilde{\mathbb{D}}_{q,p}^{n}$. Then, for
every bounded continuous function $f\colon\mathbb{R}\to\mathbb{R}$, we have

$\frac{1}{n}\sum_{i=1}^{n}f(\widetilde{X}_{i}^{(n)})\xrightarrow[n\to\infty]{\mathbb{P}}\int_{\mathbb{R}}f(x)\,\nu_{q,p}(\mathrm{d}x),$

where $\nu_{q,p}$ is a symmetric probability measure on $\mathbb{R}$
absolutely continuous with respect to Lebesgue measure and with density
function $f_{q,p}\colon\mathbb{R}\to[0,\infty)$ satisfying
$f_{q,p}(x)=\frac{1}{2}G^{\prime}(|x|)$, where
$G\colon[0,\infty)\to\mathbb{R}$ satisfies $G(0)=0$ and is the unique solution
to the differential equation

$\displaystyle G^{\prime\prime}(x)$
$\displaystyle=-G^{\prime}(x)\big{(}1-G(x)\big{)}^{p/q-1}x^{p-1},\quad
x\in(0,r_{p,q}),$

with $G(0)=0$ and $\lim_{x\uparrow r_{p,q}}G^{\prime}(x)=0$, where
$r_{p,q}\in(0,\infty]$ is the first point such that $G(r_{p,q})=1$. Further,
we conjecture that $r_{p,q}=\infty$ if and only if $p=q$.

Varying the free parameter $G^{\prime}(0)>0$, we obtain different solutions,
see Figure 1 for a simulation. If $G^{\prime}(0)$ is smaller than some
critical value $c_{p,q}$, then we conjecture that $\lim_{x\to\infty}G(x)<1$.
If $G^{\prime}(0)=c_{p,q}$, then there is a (minimal) value $r_{p,q}$ such
that $G(r_{p,q})=1$ and we have $\lim_{x\uparrow r_{p,q}}G^{\prime}(x)=0$.
This critical solution is the one that appears in the above conjecture.

Figure 1: Simulation of solutions of the differential equation in Conjecture 1
with different values of $G^{\prime}(0)$.

######  Remark 5.

The conjecture encapsulates the case $p=q$ for $\ell_{p}^{n}$ balls, where it
correctly returns $p$-Gaussian densities, and the case $p=1$, where it gives
the limiting distribution obtained in Theorem B.

##  3 Proofs of the main results

We shall now present the proofs of our main results presented in Section 2. In
what follows, given a set $A\subseteq\mathbb{R}^{n}$, we shall denote by ${\rm
conv}(A)$ the convex hull of the set $A$ and by $\partial A$ its boundary.
Moreover, we denote by $e_{1},\dots,e_{n}$ the standard unit vectors in
$\mathbb{R}^{n}$.

###  3.1 Proof of Theorem A

Note that due to the 1-symmetry of Lorentz norms (and thus unit balls) it is
sufficient to look at the Weyl chamber

$W:=\big{\\{}x\in\mathbb{R}^{n}\colon x_{1}\geq\cdots\geq x_{n}\geq
0\big{\\}}$ (4)

intersected with $\mathbb{B}_{q,1}^{n}$. This set then takes the form

$\mathbb{B}_{q,1}^{n}\cap W=\Big{\\{}x\in
W\colon\sum_{i=1}^{n}i^{1/q-1}x_{i}\leq 1\Big{\\}}.$

In a first step, we determine the extreme points (vertices) of the polytope
$\mathbb{B}_{q,1}^{n}\cap W$.

######  Lemma 4.

Let $1<q\leq\infty$. The extreme points of $\mathbb{B}_{q,1}^{n}\cap W$ are
given by

$0,Me_{1},Me_{2},\dots,Me_{n},$

where

$M:=\begin{pmatrix}\kappa_{q}(1)^{-1}&\kappa_{q}(2)^{-1}&\cdots&\kappa_{q}(n)^{-1}\\\
0&\kappa_{q}(2)^{-1}&\cdots&\kappa_{q}(n)^{-1}\\\ 0&0&\ddots&\vdots\\\
0&0&\cdots&\kappa_{q}(n)^{-1}\\\
\end{pmatrix}\quad\text{with}\quad\kappa_{q}(j)=\sum_{i=1}^{j}i^{1/q-1},\quad
j\in\\{1,\dots,n\\}.$

###### Proof.

First, we note that the Weyl chamber $W$ is the positive hull of the summing
basis

$Ae_{1}=e_{1},Ae_{2}=e_{1}+e_{2},\dots,Ae_{n}=e_{1}+\cdots+e_{n}\quad\text{with}\quad
A=\begin{pmatrix}1&1&\cdots&1\\\ 0&1&\cdots&1\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&1\\\ \end{pmatrix},$

because every $x=(x_{i})_{i=1}^{n}\in W$ can be represented as

$x=\sum_{i=1}^{n}(x_{i}-x_{i+1})(Ae_{i})=A\Big{(}\sum_{i=1}^{n}(x_{i}-x_{i+1})e_{i}\Big{)},$

where we set $x_{n+1}:=0$. A substitution gives

$\mathbb{B}_{q,1}^{n}\cap
W=\Big{\\{}x\in\mathbb{R}^{n}\colon\sum_{i=1}^{n}i^{1/q-1}x_{i}\leq
1,x=A\Big{(}\sum_{i=1}^{n}y_{i}e_{i}\Big{)},y_{i}\geq
0\Big{\\}}=\Big{\\{}Ay\in\mathbb{R}^{n}\colon\sum_{i=1}^{n}\kappa_{q}(i)y_{i}\leq
1,y_{i}\geq 0\Big{\\}}.$

Since $A$ is linear and bijective, it follows that the extreme points are
given by $0$ and $A(\kappa_{q}(i)^{-1}e_{i})=Me_{i}$, $i\in\\{1,\dots,n\\}$. ∎

We now use this lemma to prove Theorem A.

###### Proof of Theorem A.

From Lemma 4, we know the extreme points of $\mathbb{B}_{q,1}^{n}\cap W$,
which are given by the images under $M$ of $0$ and the unit vectors
$e_{1},\dots,e_{n}$; $M$ is given as in Lemma 4. Thus

$\mathbb{B}_{q,1}^{n}\cap W={\rm
conv}\big{\\{}0,Me_{1},\dots,Me_{n}\big{\\}}=M\big{(}{\rm
conv}\big{\\{}0,e_{1},\dots,e_{n}\big{\\}}\big{)}.$ (5)

The convex hull ${\rm conv}\big{\\{}0,e_{1},\dots,e_{n}\big{\\}}$ is the
projection of the standard $n$-dimensional simplex $\Delta_{n}$ in
$\mathbb{R}^{n+1}$, i.e., of

$\Delta_{n}:=\Bigg{\\{}x=(x_{i})_{i=1}^{n+1}\in\mathbb{R}^{n+1}\,:\,x_{i}\geq
0,\,\sum_{i=1}^{n+1}x_{i}=1\Big{\\}}={\rm
conv}\big{\\{}e_{1},\dots,e_{n+1}\bigg{\\}}\subseteq\mathbb{R}^{n+1},$

onto the first $n$ coordinates. It is well known (see, e.g., [44]) that

$Z^{(n)}:=\Big{(}\frac{E_{1}}{\sum_{j=1}^{n+1}E_{j}},\dots,\frac{E_{n+1}}{\sum_{j=1}^{n+1}E_{j}}\Big{)}$

is uniformly distributed on $\Delta_{n}$, where $E_{1},\dots,E_{n+1}$ are
i.i.d. standard exponential random variables. Applying the projection and then
$M$ to $Z^{(n)}$, we see that by (5) the random vector

$\Big{(}\frac{\sum_{j=1}^{n}\kappa_{q}(j)^{-1}E_{j}}{\sum_{j=1}^{n+1}E_{j}},\frac{\sum_{j=2}^{n}\kappa_{q}(j)^{-1}E_{j}}{\sum_{j=1}^{n+1}E_{j}},\dots,\frac{\sum_{j=n}^{n}\kappa_{q}(j)^{-1}E_{j}}{\sum_{j=1}^{n+1}E_{j}}\Big{)}$

is uniformly distributed on the intersection $\mathbb{B}_{q,1}^{n}\cap W$.
Using the 1-symmetry of $\mathbb{B}_{q,1}^{n}$, we may change the order of
coordinates (introducing a random permutation $\pi$ of $\\{1,\dots,n\\}$) as
well as the signs of coordinates (introducing a random sign vector
$(\varepsilon_{1},\dots,\varepsilon_{n})\in\\{-1,1\\}^{n}$) and obtain the
desired probabilistic representation. ∎

We now present the proof of the asymptotics of the volume radius of a Lorentz
ball. For this and also other proofs we shall need the following well-known
asymptotics, see for example [30, Lemma 3.4].

######  Lemma 5.

Let $\alpha>-1$. There exists a constant $c_{\alpha}\in(0,\infty)$ and an
absolute constant $c\in(0,\infty)$ such that for all $n\in\mathbb{N}$,

$\Big{|}\sum_{i=1}^{n}i^{\alpha}-\frac{1}{\alpha+1}n^{\alpha+1}\Big{|}\leq
c_{\alpha}n^{\max\\{0,\alpha\\}}\qquad\text{ and
}\qquad\Big{|}\sum_{i=1}^{n}i^{-1}-\log n\Big{|}\leq c.$

###### Proof of Corollary 1.

From Lemma 4, see Equation (5), we know that $\mathbb{B}_{q,1}^{n}\cap
W=M({\rm conv}\\{0,e_{1},\dots,e_{n}\\})$. Together with the $1$-symmetry,
this leads to

$\displaystyle\mathrm{vol}_{n}(\mathbb{B}_{q,1}^{n})$
$\displaystyle=2^{n}n!\cdot\mathrm{vol}_{n}\big{(}\mathbb{B}_{q,1}^{n}\cap
W\big{)}=2^{n}\det(M)=2^{n}\prod_{i=1}^{n}\kappa_{q}(i)^{-1},$

where $W$ is as in (4) and $M$ as in Lemma 4. Now, observe that by Lemma 5 for
a suitable constant $c_{q}\in(0,\infty)$ just depending on the parameter $q>1$
the following asymptotics hold:

$|\kappa_{q}(i)-qi^{1/q}|\leq c_{q}\,\,\text{ if
}q<\infty\quad\text{and}\quad|\kappa_{q}(i)-\log(i+1)|\leq
c_{\infty}\,\,\text{ if }q=\infty.$ (6)

We start with case $q<\infty$. Here, we have

$\displaystyle\log\Big{(}\prod_{i=1}^{n}\kappa_{q}(i)^{-1}\Big{)}^{1/n}$
$\displaystyle=-\frac{1}{n}\sum_{i=1}^{n}\log\kappa_{q}(i)=-\frac{1}{n}\sum_{i=1}^{n}\log\Big{(}\frac{\kappa_{q}(i)\cdot
qi^{1/q}}{qi^{1/q}}\Big{)}$ (7) $\displaystyle=-\log
q-\frac{1}{qn}\sum_{i=1}^{n}\log
i-\frac{1}{n}\sum_{i=1}^{n}\log\Big{(}\frac{\kappa_{q}(i)}{qi^{1/q}}\Big{)}.$
(8)

Using $\log(n!)=n\log n-n+O(\log n)$, we deduce that the first sum on the
right-hand side satisfies, as $n\to\infty$,

$-\frac{1}{qn}\sum_{i=1}^{n}\log
i=-\frac{1}{q}\log(n)+\frac{1}{q}+O\Big{(}\frac{\log n}{n}\Big{)}.$

By (6) we have

$\lim_{i\to\infty}\log\Big{(}\frac{\kappa_{q}(i)}{qi^{1/q}}\Big{)}=0.$

By Cesàro’s lemma,

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\log\Big{(}\frac{\kappa_{q}(i)}{qi^{1/q}}\Big{)}=0.$

This shows the desired result. If $q=\infty$, we proceed similarly but now use

$\frac{1}{n}\sum_{i=3}^{n}\log\log i=\frac{1}{n}\int_{3}^{n}\log\log
x+O\Big{(}\frac{1}{\log n}\Big{)}=\log\log n-\frac{{\rm
li}(n)}{n}+O\Big{(}\frac{1}{\log n}\Big{)}=\log\log n+O\Big{(}\frac{1}{\log
n}\Big{)},$

where the asymptotics of the logarithmic integral ${\rm
li}(x):=\int_{0}^{x}\frac{\mathrm{d}t}{\ln(t)},x>1,$ are well known ([1,
Chapter 5]). ∎

###  3.2 Proof of Theorems B

We now continue with the proof of Theorem B, i.e., the weak convergence
results of the empirical distribution. In what follows, for two sequences
$(a_{n})_{n\in\mathbb{N}}$ and $(b_{n})_{n\in\mathbb{N}}$ of non-negative real
numbers, we shall write $a_{n}\lesssim b_{n}$ if there exists $C\in(0,\infty)$
such that for all $n\in\mathbb{N}$ it holds that $a_{n}\leq Cb_{n}$. If in
addition $a_{n}\gtrsim b_{n}$, then we write $a_{n}\asymp b_{n}$. The
following statement will be crucial in the proof.

######  Lemma 6.

Let $1<q\leq\infty$ and for $n\in\mathbb{N}$ assume that
$\widetilde{X}^{(n)}=(\widetilde{X}_{1}^{(n)},\dots,\widetilde{X}_{n}^{(n)})$
is uniformly distributed in $\widetilde{\mathbb{D}}_{q,1}^{n}$. Then, for any
sequence $(a_{n})_{n\in\mathbb{N}}$ with $a_{n}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\infty$, we have

$\mathbb{P}\Bigg{[}\sup_{1\leq i\leq
n}\Big{|}(\widetilde{X}_{i}^{(n)})^{*}-G_{q}\Big{(}\frac{i-1}{n}\Big{)}\Big{|}\geq
a_{n}\delta_{n}\Bigg{]}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0,$

where

$G_{q}\colon[0,1]\to[0,(q-1)^{-1}],\quad x\mapsto
G_{q}(x)=\frac{1}{q-1}(1-x^{1-1/q})\quad\text{if }q<\infty$

and $G_{\infty}\colon[0,1]\to[0,1]$, $x\mapsto 1-x$. The error bounds may be
chosen as follows:

$\delta_{n}=\begin{cases}n^{-(1-1/q)}&:1<q<2,\\\ n^{-1/2}\log n&:q=2,\\\
n^{-1/q}&:q\in(2,\infty),\\\ (\log n)^{-1}&:q=\infty,\\\
\end{cases}\quad\text{where }n\in\mathbb{N}.$

###### Proof.

Let $1<q<\infty$ and $i\in\\{1,\dots,n\\}$. Using the probabilistic
representation from Theorem A, we obtain for the $i^{th}$-largest entry
$(\widetilde{X}_{i}^{(n)})^{*}$ of $(|\widetilde{X}_{i}^{(n)}|)_{i=1}^{n}$
that

$(\widetilde{X}_{i}^{(n)})^{*}\stackrel{{\scriptstyle{\rm
d}}}{{=}}Y_{i}^{(n)}\Big{(}\frac{1}{n}\sum_{j=1}^{n+1}E_{j}\Big{)}^{-1}\quad\text{with}\quad
Y_{i}^{(n)}=\frac{1}{n}\sum_{j=i}^{n}\frac{n^{1/q}}{\kappa_{q}(j)}E_{j},$

and the same holds in the sense of joint distributions. We will prove the
statement for the random variable $Y_{i}^{(n)}$ instead of
$(\widetilde{X}_{i}^{(n)})^{*}$ and then the statement of the lemma easily
follows, because as a consequence of the triangle inequality and the sub-
additivity of the supremum, we have

$\displaystyle\mathbb{P}\Bigg{[}\sup_{1\leq i\leq
n}\Big{|}(\widetilde{X}_{i}^{(n)})^{*}-G_{q}\Big{(}\frac{i-1}{n}\Big{)}\Big{|}\geq
a_{n}\delta_{n}\Bigg{]}$ $\displaystyle\leq\mathbb{P}\Bigg{[}\sup_{1\leq i\leq
n}\Big{|}(\widetilde{X}_{i}^{(n)})^{*}-Y_{i}^{(n)}\Big{|}\geq\frac{a_{n}}{2}\delta_{n}\Bigg{]}+\mathbb{P}\Bigg{[}\sup_{1\leq
i\leq
n}\Big{|}Y_{i}^{(n)}-G_{q}\Big{(}\frac{i-1}{n}\Big{)}\Big{|}\geq\frac{a_{n}}{2}\delta_{n}\Bigg{]}.$

In order to estimate the first probability on the right-hand side, we make use
of the fact that

$\sup_{1\leq i\leq
n}\Big{|}(\widetilde{X}_{i}^{(n)})^{*}-Y_{i}^{(n)}\Big{|}\stackrel{{\scriptstyle{\rm
d}}}{{=}}\big{(}\sup_{1\leq i\leq
n}Y_{i}^{(n)}\big{)}\cdot\Big{|}\Big{(}\frac{1}{n}\sum_{j=1}^{n+1}E_{j}\Big{)}^{-1}-1\Big{|}=\|Y^{(n)}\|_{\infty}\cdot\Big{|}\Big{(}\frac{1}{n}\sum_{j=1}^{n+1}E_{j}\Big{)}^{-1}-1\Big{|}.$
(9)

From the proof below it will follow that the first factor on the right-hand
side of (9) is bounded by a suitable constant with probability tending to one.
The second factor satisfies (via the central limit theorem)

$\displaystyle\mathbb{P}\Bigg{[}\Big{|}\Big{(}\frac{1}{n}\sum_{j=1}^{n+1}E_{j}\Big{)}^{-1}-1\Big{|}\geq\delta_{n}\Bigg{]}$
$\displaystyle\leq\mathbb{P}\Bigg{[}\frac{1}{n}\sum_{j=1}^{n+1}E_{j}\leq(1+\delta_{n})^{-1}\Bigg{]}\leq\mathbb{P}\Bigg{[}\frac{1}{n}\sum_{j=1}^{n}E_{j}\leq(1+\delta_{n})^{-1}\Bigg{]}$
$\displaystyle=\mathbb{P}\Bigg{[}\frac{s_{n}}{n}\sum_{j=1}^{n}(E_{j}-1)\leq-1\Bigg{]}=\mathbb{P}\Bigg{[}\frac{1}{\sqrt{n}}\sum_{j=1}^{n}(E_{j}-1)\leq-\frac{\sqrt{n}}{s_{n}}\Bigg{]}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0,$

where $s_{n}:=1+\delta_{n}^{-1}\asymp\delta_{n}^{-1}$ satisfies
$\sqrt{n}/s_{n}\to+\infty$, as a consequence of the definition of $\delta_{n}$
stated in the lemma. Taking into account (9) and the upper bound on the
largest coordinate of $Y^{(n)}$, one easily derives the statement for
$\widetilde{X}^{(n)}$ instead of $Y^{(n)}$.

We now start with the proof of the statement for $Y_{i}^{(n)}$. Consider the
decomposition

$Y^{(n)}_{i}-G_{q}\Big{(}\frac{i-1}{n}\Big{)}=\frac{1}{n}\sum_{j=i}^{n}\frac{n^{1/q}}{\kappa_{q}(j)}E_{j}-\frac{1}{q}\int_{(i-1)/n}^{1}x^{-1/q}\,\mathrm{d}x=\frac{1}{n}\sum_{j=i}^{n}A_{j}+\frac{1}{n}\sum_{j=i}^{n}B_{j}+\frac{1}{qn}\sum_{j=i}^{n}C_{j}$
(10)

with summands

$A_{j}=\frac{n^{1/q}}{\kappa_{q}(j)}(E_{j}-1),\quad
B_{j}=\frac{n^{1/q}}{\kappa_{q}(j)}-\frac{n^{1/q}}{qj^{1/q}},\quad
C_{j}=\Big{(}\frac{j}{n}\Big{)}^{-1/q}-\int_{j-1}^{j}\Big{(}\frac{x}{n}\Big{)}^{-1/q}\,\mathrm{d}x,\quad
j=i,\dots,n.$

Here we used $G_{q}(x)=\frac{1}{q}\int_{x}^{1}y^{-1/q}\,\mathrm{d}y$,
$x\in[0,1]$. Only the first sum on the right-hand side of (10) is random and
its variance is

$\frac{1}{n^{2}}\sum_{j=i}^{n}{\rm
Var}(A_{j})=\frac{1}{n^{2}}\sum_{j=i}^{n}\Big{(}\frac{n^{1/q}}{\kappa_{q}(j)}\Big{)}^{2}\lesssim_{q}n^{-2(1-1/q)}\sum_{j=1}^{n}j^{-2/q}\lesssim_{q}\begin{cases}n^{-2(1-1/q)}&\text{if
}1<q<2,\\\ n^{-1}\log n&\text{if }q=2,\\\ n^{-1}&\text{if }q>2,\\\
\end{cases}$

which tends to zero as $n\to\infty$. Here we used the asymptotics
$\kappa_{q}(j)\asymp_{q}j^{1/q}$ and sum-integral approximation, see (6) and
Lemma 5.

For the second sum we have due to (6)

$|B_{j}|=\frac{n^{1/q}}{qj^{1/q}}\frac{|qj^{1/q}-\kappa_{q}(j)|}{\kappa_{q}(j)}\asymp_{q}\frac{n^{1/q}}{j^{1/q}}\frac{1}{j^{1/q}}=n^{1/q}j^{-2/q},$

which gives that

$\frac{1}{n}\sum_{j=i}^{n}|B_{j}|\lesssim_{q}n^{1/q-1}\sum_{j=1}^{n}j^{-2/q}\lesssim_{q}\begin{cases}n^{-(1-1/q)}&\text{if
}1<q<2,\\\ n^{-1/2}\log n&\text{if }q=2,\\\ n^{-1/q}&\text{if }q>2.\\\
\end{cases}$

For the third sum we use that for any $j\geq 1$, we find $\xi\in(j-1,j)$ such
that

$0\leq\int_{j-1}^{j}\Big{(}\frac{x}{n}\Big{)}^{-1/q}\mathrm{d}x-\Big{(}\frac{j}{n}\Big{)}^{-1/q}=\Big{(}\frac{\xi}{n}\Big{)}^{-1/q}-\Big{(}\frac{j}{n}\Big{)}^{-1/q}\lesssim\Big{(}\frac{j}{n}\Big{)}^{-1/q}\frac{1}{j},$

which follows from the mean value theorem used twice. Thus, the third sum
satisfies

$\frac{1}{n}\sum_{j=i}^{n}|C_{j}|\lesssim_{q}n^{-(1-1/q)}\sum_{j=1}^{n}j^{-(1+1/q)}\lesssim_{q}n^{-1+1/q}.$

Lastly, we note that

$\displaystyle\mathbb{P}\Bigg{[}\sup_{1\leq i\leq
n}\Big{|}Y_{i}^{(n)}-G_{q}\Big{(}\frac{i-1}{n}\Big{)}\Big{|}\geq
a_{n}\delta_{n}\Bigg{]}$ $\displaystyle\leq\mathbb{P}\Bigg{[}\sup_{1\leq i\leq
n}\Big{|}\frac{1}{n}\sum_{j=i}^{n}A_{j}\Big{|}\geq\frac{a_{n}}{3}\delta_{n}\Bigg{]}$
$\displaystyle+\mathbb{P}\Bigg{[}\sup_{1\leq i\leq
n}\Big{|}\frac{1}{n}\sum_{j=i}^{n}B_{j}\Big{|}\geq\frac{a_{n}}{3}\delta_{n}\Bigg{]}$
$\displaystyle+\mathbb{P}\Bigg{[}\sup_{1\leq i\leq
n}\Big{|}\frac{1}{n}\sum_{j=i}^{n}C_{j}\Big{|}\geq\frac{a_{n}}{3}\delta_{n}\Bigg{]}.$

For large $n$ the last two summands vanish because of the previous estimates
and the first tends to zero due to Kolmogorov’s inequality ([37, Thm. 5.28]).
This proves the statement for $q<\infty$.

If $q=\infty$ we need to carry out several modifications, in particular adapt
the definition of $Y^{(n)}$ by replacing $n^{1/q}$ with $\log(n+1)$. Let us
write for $i\in\\{1,\dots,n\\}$ again the coordinate as in equation (10),
i.e.,

$Y^{(n)}_{i}-G_{\infty}\Big{(}\frac{i-1}{n}\Big{)}=\frac{1}{n}\sum_{j=i}^{n}\frac{\log(n+1)}{\kappa_{\infty}(j)}E_{j}-\Big{(}1-\frac{i-1}{n}\Big{)}=\frac{1}{n}\sum_{j=i}^{n}A_{j}+\frac{1}{n}\sum_{j=i}^{n}B_{j}+\frac{1}{n}\sum_{j=i}^{n}C_{j}$

with summands

$A_{j}=\frac{\log(n+1)}{\kappa_{\infty}(j)}(E_{j}-1),\quad
B_{j}=\frac{\log(n+1)}{\kappa_{\infty}(j)}-\frac{\log(n+1)}{\log(j+1)},\quad
C_{j}=\frac{\log(n+1)}{\log(j+1)}-1.$

One can estimate the errors similarly after adjusting $\delta_{n}$. For the
first sum we have $\kappa_{\infty}(j)\asymp\log(j+1)$ using (6) and thus

$\frac{1}{n^{2}}\sum_{j=i}^{n}{\rm
Var}(A_{j})=\frac{1}{n^{2}}\sum_{j=i}^{n}\Big{(}\frac{\log(n+1)}{\kappa_{\infty}(j)}\Big{)}^{2}\lesssim\frac{\log^{2}(n+1)}{n^{2}}\sum_{j=1}^{n}\frac{1}{\log^{2}(j+1)}\lesssim\frac{1}{n},$

where in the last estimate we used that $\log^{2}(n+1)$ is slowly varying and
thus the arithmetic mean is equivalent to the largest summand, see also [10,
Prop. 1.5.8]. Similarly, we have

$\frac{1}{n}\sum_{j=i}^{n}|B_{j}|\lesssim\frac{\log(n+1)}{n}\sum_{j=1}^{n}\frac{1}{\log^{2}(j+1)}\lesssim\frac{1}{\log(n+1)}.$

The third sum satisfies

$\frac{1}{n}\sum_{j=i}^{n}|C_{j}|\leq\frac{1}{n}\sum_{j=1}^{n}|C_{j}|=\frac{\log(n+1)}{n}\sum_{j=1}^{n}\frac{1}{\log(j+1)}-1\lesssim\frac{1}{\log(n+1)},$

where we used the approximation

$\sum_{j=1}^{n}\frac{1}{\log(j+1)}=\int_{2}^{n+1}\frac{1}{\log
x}\,\mathrm{d}x+O(1)={\rm
li}(n+1)+O(1)=\frac{n}{\log(n+1)}+O\Big{(}\frac{n}{\log^{2}(n+1)}\Big{)}.$

The proof is now completed as in the case $q<\infty$. ∎

For the proof of Theorem B we use a symmetrization trick, which we present in
the following lemma; the result may well be known, but we were unable to find
it in the literature.

######  Lemma 7.

For each $n\in\mathbb{N}$ let $X^{(n)}=(X_{1}^{(n)},\dots,X_{n}^{(n)})$ be a
random vector in $\mathbb{R}^{n}$ with identically distributed coordinates
such that
$(\varepsilon_{1}X_{1}^{(n)},\dots,\varepsilon_{n}X_{n}^{(n)})\overset{{\rm
d}}{=}X^{(n)}$ for any choice of signs
$(\varepsilon_{1},\dots,\varepsilon_{n})\in\\{-1,1\\}^{n}$. If for every
bounded continuous function $f\colon[0,\infty)\to\mathbb{R}$

$\frac{1}{n}\sum_{i=1}^{n}f\big{(}|X^{(n)}_{i}|\big{)}\xrightarrow[n\to\infty]{\mathbb{P}}\int_{0}^{\infty}f(x)\varphi(x)\,\mathrm{d}x,$

holds for a density $\varphi\colon[0,\infty)\to[0,\infty)$, then for every
bounded continuous function $f\colon\mathbb{R}\to\mathbb{R}$

$\frac{1}{n}\sum_{i=1}^{n}f\big{(}X^{(n)}_{i}\big{)}\xrightarrow[n\to\infty]{\mathbb{P}}\frac{1}{2}\int_{-\infty}^{\infty}f(x)\varphi(|x|)\,\mathrm{d}x.$

###### Proof.

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a bounded and continuous function. The
two mappings

$f_{1}\colon[0,\infty)\to\mathbb{R},\quad x\mapsto f(x)\quad\text{and}\quad
f_{-1}\colon[0,\infty)\to\mathbb{R},\quad x\mapsto f(-x)$

are again bounded and continuous on $[0,\infty)$. Let
$\varepsilon_{1},\dots,\varepsilon_{n}$ be independent Rademacher random
variables, independent of all other random objects. Then the symmetry of
$X^{(n)}$ implies

$S_{n}:=\frac{1}{n}\sum_{i=1}^{n}f\big{(}X^{(n)}_{i}\big{)}\overset{\rm
d}{=}\frac{1}{n}\sum_{i=1}^{n}f\big{(}\varepsilon_{i}|X^{(n)}_{i}|\big{)}=\frac{1}{n}\sum_{i=1}^{n}f_{\varepsilon_{i}}\big{(}|X^{(n)}_{i}|\big{)}.$

By linearity of the conditional expectation, we obtain

$\displaystyle\mathbb{E}\big{[}S_{n}\big{|}X^{(n)}\big{]}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\big{[}f_{\varepsilon_{i}}\big{(}|X^{(n)}_{i}|\big{)}\big{|}X^{(n)}\big{]}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\frac{(f_{1}+f_{-1})(|X^{(n)}_{i}|)}{2}\xrightarrow[n\to\infty]{\mathbb{P}}\int_{0}^{\infty}\frac{(f_{1}+f_{-1})(x)}{2}\varphi(x)\,\mathrm{d}x=\frac{1}{2}\int_{-\infty}^{\infty}f(x)\varphi(|x|)\,\mathrm{d}x.$
(11)

Further, the conditional variance satisfies

${\rm
Var}\big{[}S_{n}\big{|}X^{(n)}\big{]}=\frac{1}{2n^{2}}\sum_{i=1}^{n}\bigg{(}f_{1}\big{(}|X_{i}^{(n)}|\big{)}-\frac{(f_{1}+f_{-1})(|X_{i}^{(n)}|)}{2}\bigg{)}^{2}+\bigg{(}f_{-1}\big{(}|X_{i}^{(n)}|\big{)}-\frac{(f_{1}+f_{-1})(|X_{i}^{(n)}|)}{2}\bigg{)}^{2}\leq\frac{4\|f\|_{\infty}^{2}}{n}.$

Conditioned on $X^{(n)}$, Chebyshev’s inequality gives, for every
$\varepsilon>0$,

$\mathbb{P}\big{[}|S_{n}-\mathbb{E}[S_{n}\big{|}X^{(n)}]|>\varepsilon\big{|}X^{(n)}\big{]}\leq\frac{{\rm
Var}\big{[}S_{n}\big{|}X^{(n)}\big{]}}{\varepsilon^{2}}.$

Thus, the law of total expectation yields

$\mathbb{P}\big{[}|S_{n}-\mathbb{E}[S_{n}\big{|}X^{(n)}]|>\varepsilon\big{]}\leq\frac{\mathbb{E}\big{[}\,{\rm
Var}[S_{n}\big{|}X^{(n)}]\big{]}}{\varepsilon^{2}}\leq\frac{4\|f\|_{\infty}^{2}}{\varepsilon^{2}n}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0.$

This together with (3.2) implies

$S_{n}\xrightarrow[n\to\infty]{\mathbb{P}}\frac{1}{2}\int_{-\infty}^{\infty}f(x)\varphi(|x|)\,\mathrm{d}x,$

which completes the proof. ∎

###### Proof of Theorem B.

We use Lemma 7, which allows us to pass to the absolute values. Let
$\varepsilon>0$ and $f\colon[0,\infty)\to[0,\infty)$ be bounded and
continuous. We have

$\displaystyle\mathbb{P}\Bigg{[}\Big{|}\frac{1}{n}\sum_{i=1}^{n}f\big{(}|\widetilde{X}_{i}^{(n)}|\big{)}-\int_{\mathbb{R}_{+}}f(x)\,\nu_{q,1}(\mathrm{d}x)\Big{|}>\varepsilon\Bigg{]}$
$\displaystyle\leq\mathbb{P}\Bigg{[}\Big{|}\frac{1}{n}\sum_{i=1}^{n}f\big{(}|\widetilde{X}_{i}^{(n)}|\big{)}-\frac{1}{n}\sum_{i=1}^{n}f\Big{(}G_{q}\Big{(}\frac{i-1}{n}\Big{)}\Big{)}\Big{|}>\frac{\varepsilon}{2}\Bigg{]}$
$\displaystyle+\mathbb{P}\Bigg{[}\Big{|}\frac{1}{n}\sum_{i=1}^{n}f\Big{(}G_{q}\Big{(}\frac{i-1}{n}\Big{)}\Big{)}-\int_{\mathbb{R}_{+}}f(x)\,\nu_{q,1}(\mathrm{d}x)\Big{|}>\frac{\varepsilon}{2}\Bigg{]}.$
(12)

For the first summand on the right-hand side, we note that

$\Big{|}\frac{1}{n}\sum_{i=1}^{n}f\big{(}|\widetilde{X}_{i}^{(n)}|\big{)}-\frac{1}{n}\sum_{i=1}^{n}f\Big{(}G_{q}\Big{(}\frac{i-1}{n}\Big{)}\Big{)}\Big{|}\leq\sup_{1\leq
i\leq
n}\Big{|}f\big{(}(\widetilde{X}_{i}^{(n)})^{*}\big{)}-f\Big{(}G_{q}\Big{(}\frac{i-1}{n}\Big{)}\Big{)}\Big{|},$

since we may arrange the absolute values in any order without changing the
average. Let $(a_{n})_{n\in\mathbb{N}}$ be a sequence with
$a_{n}\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}\infty$ such that
$a_{n}\delta_{n}\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}0$ and
$a_{n}\delta_{n}\leq 1$ for all $n\in\mathbb{N}$. For $n\in\mathbb{N}$, we
define the event

$A_{n}:=\Bigg{\\{}\sup_{1\leq i\leq
n}\Big{|}(\widetilde{X}_{i}^{(n)})^{*}-G_{q}\Big{(}\frac{i-1}{n}\Big{)}\Big{|}\leq
a_{n}\delta_{n}\Bigg{\\}}.$

Then on $A_{n}$ it holds that:
$(\widetilde{X}_{i}^{(n)})^{*}\leq(\widetilde{X}_{1}^{(n)})^{*}\leq
G_{q}(0)+1$ for all $i\in\\{1,\dots,n\\}$ and

$\sup_{1\leq i\leq
n}\Big{|}f\big{(}(\widetilde{X}_{i}^{(n)})^{*}\big{)}-f\Big{(}G_{q}\Big{(}\frac{i-1}{n}\Big{)}\Big{)}\Big{|}\leq\sup_{\begin{subarray}{c}x,y\in[0,G_{q}(0)+1]\\\
|x-y|\leq a_{n}\delta_{n}\end{subarray}}|f(x)-f(y)|,$

which tends to zero because $f$ is uniformly continuous on $[0,G_{q}(0)+1]$.
By Lemma 6 it holds that $\mathbb{P}[A_{n}]\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}1$, and thus the first summand in (3.2) tends
to zero as well. For the second summand in (3.2), we note that the continuity
of $f$ and $G_{q}\colon[0,1]\to[0,\infty)$ imply via a substitution that

$\frac{1}{n}\sum_{i=1}^{n}f\Big{(}G_{q}\Big{(}\frac{i-1}{n}\Big{)}\Big{)}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\int_{0}^{1}f(G_{q}(x))\,\mathrm{d}x=\int_{0}^{G_{q}(0)}f(x)(-G_{q}^{-1})^{\prime}(x)\,\mathrm{d}x,$

where

$G_{q}^{-1}\colon[0,G_{q}(0)],\quad x\mapsto(1-(q-1)x)^{q/(q-1)},$

if $q<\infty$ and $G_{\infty}^{-1}=G_{\infty}$, such that
$(-G_{q}^{-1})^{\prime}$ is the density of $\nu_{q,1}$.∎

With Theorem B at our disposal, we are able to deduce Corollary 2 on the
asymptotic distribution of a single coordinate.

###### Proof of Corollary 2.

The result essentially follows from the exchangeability of the coordinates of
$\widetilde{X}^{(n)}=(\widetilde{X}_{1}^{(n)},\dots,\widetilde{X}_{n}^{(n)})$
and in mathematical physics literature is known as propagation of chaos, see,
e.g., the work [63] of Sznitman; recall that exchangeability means that any
permutation of coordinates has the same joint distribution as the original
one. For convenience of the reader, we adapt an argument from [15, p. 326].

Let $k\in\mathbb{N}$. We show for any bounded and continuous function
$f\colon\mathbb{R}^{k}\to\mathbb{R}$ that

$\mathbb{E}\big{[}f(\widetilde{X}_{1}^{(n)},\dots,\widetilde{X}_{k}^{(n)})\big{]}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\big{(}\mathbb{E}f(Y)\big{)}^{k},$

where $Y$ is real random variable distributed according to the Lebesgue
density $f_{q,1}$. It is sufficient to assume that $f=\prod_{i=1}^{k}f_{i}$
for $f_{i}\colon\mathbb{R}\to\mathbb{R}$ bounded and continuous (see [15,
Appendix D, p. 356]). Due to exchangeability and linearity, we have

$\mathbb{E}\Big{[}\prod_{i=1}^{k}f_{i}(\widetilde{X}_{i}^{(n)})\Big{]}=\mathbb{E}\Bigg{[}\frac{(n-k)!}{n!}\sum_{i_{1}\neq\cdots\neq
i_{k}}\prod_{i=1}^{k}f_{i}(\widetilde{X}_{i_{j}}^{(n)})\Bigg{]}.$

Let
$L_{\widetilde{X}^{(n)}}=\frac{1}{n}\sum_{i=1}^{n}\delta_{\widetilde{X}^{(n)}_{i}}$.
Then we also have that

$\mathbb{E}\Bigg{[}\int\prod_{i=1}^{k}f_{i}\,\mathrm{d}L_{\widetilde{X}^{(n)}}^{\otimes
k}\Bigg{]}=\mathbb{E}\Bigg{[}\prod_{i=1}^{k}\frac{1}{n}\sum_{j=1}^{n}f_{i}(\widetilde{X}_{j}^{(n)})\Bigg{]}=\frac{1}{n^{k}}\mathbb{E}\Bigg{[}\sum_{i_{1},\dots,i_{k}}\prod_{i=1}^{k}f_{i}(\widetilde{X}_{i_{j}}^{(n)})\Bigg{]}.$

Therefore, it holds that

$\Big{|}\mathbb{E}\prod_{i=1}^{k}f_{i}(\widetilde{X}_{i}^{(n)})-\mathbb{E}\int\prod_{i=1}^{k}f_{i}\mathrm{d}L_{\widetilde{X}^{(n)}}^{\otimes
k}\Big{|}=\Big{|}1-\frac{n!}{(n-k)!n^{k}}\Big{|}\max_{1\leq i\leq
k}\|f_{i}\|_{\infty}^{k}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0.$

Applied to the $f_{i}$’s, Theorem B and the continuous mapping theorem give
the convergence

$\int\prod_{i=1}^{k}f_{i}\,\mathrm{d}L_{\widetilde{X}^{(n)}}^{\otimes
k}=\prod_{i=1}^{k}\int
f_{i}\,\mathrm{d}L_{\widetilde{X}^{(n)}}\xrightarrow[n\to\infty]{\mathbb{P}}\prod_{i=1}^{k}f_{i}(Y).$

By uniform boundedness we can take expectations on both sides and this
completes the proof. ∎

###  3.3 Proof of Theorem C

We now prove the central limit theorem for the maximum norm of a random vector
in a Lorentz ball.

###### Proof of Theorem C.

We have

$\|\widetilde{X}^{(n)}\|_{\infty}\overset{{\rm
d}}{=}\frac{Y_{n}}{Z_{n}}\quad\text{with}\quad
Y_{n}:=\frac{1}{n}\sum_{j=1}^{n}\frac{n^{1/q}}{\kappa_{q}(j)}E_{j}\quad\text{and}\quad
Z_{n}:=\frac{1}{n}\sum_{j=1}^{n+1}E_{j}.$

We first prove (i). Let $1\leq q<2$ and write

$Y_{n}=n^{1/q-1}\sum_{j=1}^{n}\frac{E_{j}-1}{\kappa_{q}(j)}+\mu_{q,n}\quad\text{with}\quad\mu_{q,n}=\frac{1}{n}\sum_{j=1}^{n}\frac{n^{1/q}}{\kappa_{q}(j)}$

as in the statement of the theorem. Note that

$\mathbb{E}\Big{[}\frac{E_{j}-1}{\kappa_{q}(j)}\Big{]}=0\quad\text{and}\quad\mathrm{Var}\Big{[}\frac{E_{j}-1}{\kappa_{q}(j)}\Big{]}=\frac{1}{\kappa_{q}(j)^{2}}.$

Since $\kappa_{q}(j)\asymp_{q}j^{1/q}$, we have

$\sum_{j=1}^{\infty}\mathrm{Var}\Big{[}\frac{E_{j}-1}{\kappa_{q}(j)}\Big{]}=\sum_{j=1}^{\infty}\frac{1}{\kappa_{q}(j)^{2}}<\infty$

and thus by the martingale convergence theorem ([37, Thm. 11.4])

$\widetilde{Y}_{n}:=n^{1-1/q}Y_{n}-\sum_{j=1}^{n}\frac{1}{\kappa_{q}(j)}=\sum_{j=1}^{n}\frac{E_{j}-1}{\kappa_{q}(j)}\xrightarrow[n\to\infty]{\rm
a.s.}\sum_{j=1}^{\infty}\frac{E_{j}-1}{\kappa_{q}(j)}=:R_{q}.$ (13)

Write

$n^{1-1/q}\big{(}\|\widetilde{X}^{(n)}\|_{\infty}-\mu_{q,n}\big{)}\overset{{\rm
d}}{=}n^{1-1/q}\Big{(}\frac{Y_{n}}{Z_{n}}-\mu_{q,n}\Big{)}=\frac{\widetilde{Y}_{n}+\sum_{j=1}^{n}\frac{1}{\kappa_{q}(j)}}{1+n^{-1/2}\widetilde{Z}_{n}+n^{-1}}-\sum_{j=1}^{n}\frac{1}{\kappa_{q}(j)},$

where
$\widetilde{Z}_{n}:=n^{-1/2}\sum_{j=1}^{n+1}(E_{j}-1)\stackrel{{\scriptstyle{\rm
d}}}{{\longrightarrow}}\mathscr{N}(0,1)$ by the central limit theorem and
Slutsky’s theorem ([37, Thm. 13.18]) since
$E_{n+1}/\sqrt{n}\stackrel{{\scriptstyle\mathbb{P}}}{{\longrightarrow}}0$.
Simplifying, by Slutsky’s theorem,

$n^{1-1/q}\big{(}\|\widetilde{X}^{(n)}\|_{\infty}-\mu_{q,n}\big{)}\overset{{\rm
d}}{=}\frac{\widetilde{Y}_{n}-n^{-1/2}\Big{(}\sum_{j=1}^{n}\frac{1}{\kappa_{q}(j)}\Big{)}(\widetilde{Z}_{n}+n^{-1/2})}{1+n^{-1/2}\widetilde{Z}_{n}+n^{-1}}\xrightarrow[n\to\infty]{{\rm
d}}R_{q},$

since
$n^{-1/2}\Big{(}\sum_{j=1}^{n}\frac{1}{\kappa_{q}(j)}\Big{)}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0$.

For convenience of the reader we prove that $R_{1}$ is Gumbel distributed
(see, e.g., [32, Theorem 1.1 (c)]). It is well-known that
$\sum_{j=1}^{n}\frac{1}{j}=\log n+\gamma+o(1)$ and from Remark 1, we deduce
that

$\sum_{j=1}^{n}\frac{E_{j}-1}{j}\overset{{\rm d}}{=}\max_{1\leq j\leq
n}E_{j}-\log n-\gamma+e_{n},$

where $e_{n}\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}0$. Now the
well-known fact that $\max_{1\leq j\leq n}E_{j}-\log
n\xrightarrow[n\to\infty]{{\rm d}}G$, where $G$ is standard Gumbel
distributed, Slutsky’s theorem and uniqueness of the limit imply that
$R_{1}\overset{{\rm d}}{=}G-\gamma$.

For the proof of (ii) let $q=2$. Defining $\widetilde{Y}_{n}$ as on the left-
hand side of (13), by a version of the Lindeberg central limit theorem from
[13, Thm. 5.3], we have

$\frac{1}{\sqrt{\log n}}\widetilde{Y}_{n}=\frac{1}{\sqrt{\log
n}}\sum_{j=1}^{n}\frac{E_{j}-1}{\kappa_{2}(j)}\xrightarrow[n\to\infty]{{\rm
d}}\mathscr{N}(0,1/4),$

since $\sum_{j=1}^{n}\frac{1}{\kappa_{2}(j)^{2}}\sim\frac{1}{4}\log
n\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}\infty$.

A similar rewriting and again Slutsky’s theorem give

$\frac{\sqrt{n}}{\sqrt{\log
n}}\big{(}\|\widetilde{X}^{(n)}\|_{\infty}-\mu_{2,n}\big{)}=\frac{\frac{\widetilde{Y}_{n}}{\sqrt{\log
n}}-\frac{n^{-1/2}}{\sqrt{\log
n}}\Big{(}\sum_{j=1}^{n}\frac{1}{\kappa_{2}(j)}\Big{)}(\widetilde{Z}_{n}+1)}{1+n^{-1/2}\widetilde{Z}_{n}+n^{-1/2}}\xrightarrow[n\to\infty]{{\rm
d}}\mathscr{N}(0,1/4),$

since $\frac{n^{-1/2}}{\sqrt{\log
n}}\Big{(}\sum_{j=1}^{n}\frac{1}{\kappa_{2}(j)}\Big{)}\asymp\frac{1}{\sqrt{\log
n}}\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}0$.

In order to prove (iii), let now $q>2$. We will show that the vector
$\sqrt{n}(Y_{n}-\mu_{q,n},Z_{n}-1)$ tends to a Gaussian random vector and then
use Taylor’s theorem to conclude the proof. We have

$\sqrt{n}(Y_{n}-\mu_{q,n},Z_{n}-1)=\frac{1}{\sqrt{n}}\sum_{j=1}^{n}\Big{(}\frac{n^{1/q}}{\kappa_{q}(j)}(E_{j}-1),E_{j}-1\Big{)}+\Big{(}0,\frac{E_{n+1}}{\sqrt{n}}\Big{)}.$

Note that, since $\frac{E_{n+1}}{\sqrt{n}}$ tends to zero in probability, by
Slutsky’s theorem we can omit the last summand as we are dealing with
convergence in distribution. For the sum of random vectors on the right-hand
side we check Lyapunov’s condition for the multivariate central limit theorem
(see, e.g., [45, Thm. 3.2.2]). To this end, let $\delta>0$ with $q>2+\delta$
and compute

$\displaystyle
n^{-(2+\delta)/2}\sum_{i=1}^{n}\mathbb{E}\Big{|}\frac{n^{1/q}}{\kappa_{q}(j)}E_{j}\Big{|}^{2+\delta}$
$\displaystyle\lesssim
n^{-(2+\delta)/2}\sum_{i=1}^{n}\Big{|}\frac{n^{1/q}}{\kappa_{q}(j)}\Big{|}^{2+\delta}$
$\displaystyle\lesssim_{q}n^{(2+\delta)(-1/2+1/q)}\sum_{i=1}^{n}j^{-(2+\delta)/q}\lesssim_{q}n^{1-(2+\delta)/2}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0.$

It remains to compute the limit of the covariance matrix. We have
$\mathrm{Var}\Big{[}\frac{1}{\sqrt{n}}\sum_{j=1}^{n}E_{j}\Big{]}=1$ for all
$n\in\mathbb{N}$ as well as

$\lim_{n\to\infty}\mathrm{Var}\Big{[}\frac{1}{\sqrt{n}}\sum_{j=1}^{n}\frac{n^{1/q}}{\kappa_{q}(j)}E_{j}\Big{]}=\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}\frac{n^{2/q}}{\kappa_{q}(j)^{2}}=\frac{1}{q^{2}}\int_{0}^{1}x^{-2/q}\mathrm{d}x=\frac{1}{q(q-2)}=:s_{q}^{2},$

where we used that $\kappa_{q}(j)\sim qj^{1/q}$ as $j\to\infty$. Similarly,

$\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}\mathrm{Cov}\Big{[}\frac{n^{1/q}}{\kappa_{q}(j)}E_{j},E_{j}\Big{]}=\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}\frac{n^{1/q}}{\kappa_{q}(j)}=\frac{1}{q}\int_{0}^{1}x^{-1/q}\mathrm{d}x=\frac{1}{q-1}=:\mu_{q,\infty}.$

Therefore, the limiting covariance matrix is

$\Sigma:=\lim_{n\to\infty}\mathrm{Cov}\big{[}\sqrt{n}(Y_{n}-\mu_{q,n},Z_{n}-1)\big{]}=\begin{pmatrix}s_{q}^{2}&\mu_{q,\infty}\\\
\mu_{q,\infty}&1\\\ \end{pmatrix}$

and we derive the central limit theorem

$\sqrt{n}(Y_{n}-\mu_{q,n},Z_{n}-1)\xrightarrow[n\to\infty]{{\rm
d}}(Y,Z)\sim\mathscr{N}(0,\Sigma).$ (14)

In order to prove a central limit theorem for $\frac{Y_{n}}{Z_{n}}$, note that
the function $(x,y)\mapsto F(x,y)=\frac{x}{y}$ is continuously differentiable
for $x,y>0$ and $\nabla F(x,y)=(\frac{1}{y},-\frac{x}{y^{2}})^{\top}$. Hence,
by Taylor’s theorem,

$\sqrt{n}\big{(}F(Y_{n},Z_{n})-F(\mu_{q,n},1)\big{)}=\sqrt{n}\big{(}(Y_{n},Z_{n})-(\mu_{q,n},1)\big{)}\nabla
F(\mu_{q,n},1)^{\top}+e_{n,q},$ (15)

where the random error term is

$e_{n,q}:=\|\sqrt{n}(Y_{n}-\mu_{q,n},Z_{n}-1)\|_{2}h(Y_{n}-\mu_{q,n},Z_{n}-1),$

where $h$ is a function tending to zero as its argument approaches zero. Due
to the CLT in (14) the random variable
$\|\sqrt{n}(Y_{n}-\mu_{q,n},Z_{n}-1)\|_{2}$ stays bounded in probability and
$h(Y_{n}-\mu_{q,n},Z_{n}-1)$ tends to zero in probability. Thus, the error
satisfies $e_{n,q}\xrightarrow[n\to\infty]{\mathbb{P}}0$. For the first term
on the right-hand side of (15), we note that $\nabla
F(\mu_{q,n},1)^{\top}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\nabla F(\mu_{q,\infty},1)^{\top}$. Therefore,
by (14) and (15) together with Slutsky’s theorem, we have

$\sqrt{n}(F(Y_{n},Z_{n})-F(\mu_{q,n},1))\xrightarrow[n\to\infty]{{\rm
d}}(Y,Z)\nabla F(\mu_{q,\infty},1)^{\top}\sim\mathscr{N}(0,\sigma_{q}^{2})$

with

$\sigma_{q}^{2}:=(1,-\mu_{q,\infty})\Sigma(1,-\mu_{q,\infty})^{\top}=\frac{1}{q(q-1)^{2}(q-2)}.$

This implies the claimed central limit theorem for
$\|\widetilde{X}^{(n)}\|_{\infty}$. ∎

###  3.4 Proof of Theorem D

We now present the proof of Theorem D, i.e., of the weak law of large numbers
for the $\ell_{r}^{n}$ norm of random points in normalized Lorentz balls. A
key ingredient is again Lemma 6.

###### Proof of Theorem D.

Let $r<\infty$ and assume first that $\widetilde{X}^{(n)}$ is uniformly
distributed on $\widetilde{\mathbb{D}}_{q,1}^{n}$. First, because of the
permutation invariance of the $\ell_{r}^{n}$ norm, we have

$n^{-1}\|\widetilde{X}^{(n)}\|_{r}^{r}=\frac{1}{n}\sum_{i=1}^{n}\Big{(}(\widetilde{X}_{i}^{(n)})^{*}\Big{)}^{r}.$

Hence, by Lemma 6 together with the same arguments as in the proof of Theorem
B,

$\Big{|}n^{-1}\|\widetilde{X}^{(n)}\|_{r}^{r}-\frac{1}{n}\sum_{i=1}^{n}G_{q}\Big{(}\frac{i-1}{n}\Big{)}^{r}\Big{|}\xrightarrow[n\to\infty]{\mathbb{P}}0.$

It is therefore sufficient to compute the deterministic limit

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}G_{q}\Big{(}\frac{i-1}{n}\Big{)}^{r}=\int_{0}^{1}G_{q}(x)^{r}\,\mathrm{d}x=\begin{cases}\int_{0}^{1}\Big{(}\frac{1}{q-1}(1-x^{1-1/q})\Big{)}^{r}\,\mathrm{d}x&\colon
q<\infty,\\\ \int_{0}^{1}(1-x)^{r}\,\mathrm{d}x&\colon q=\infty.\end{cases}$
(16)

The integral can be evaluated using a substitution and the beta function.

The case $r=\infty$ follows directly from the fact that
$\|\widetilde{X}^{(n)}\|_{\infty}=(\widetilde{X}^{(n)}_{1})^{*}$ and Lemma 6.

By Corollary 1 it remains to multiply the resulting constants by
$\frac{q}{2e^{1/q}}$ for $q<\infty$ and $\frac{1}{2}$ for $q=\infty$ in order
to obtain the result for $X^{(n)}$ uniformly distributed in
$\mathbb{D}_{q,1}$. ∎

Finally, we present the proof of the threshold result for the asymptotic
volume of intersections of normalized Lorentz and $\ell_{r}^{n}$ balls. The
proof is based on the weak law of large numbers for the $\ell_{r}^{n}$ norm
(see Theorem D). Recall that for each $r\in(0,\infty)$

$\mathrm{vol}_{n}(\mathbb{B}_{r}^{n})^{1/n}=\frac{2\Gamma(1+\frac{1}{r})}{\Gamma\big{(}1+\frac{n}{r}\big{)}^{1/n}}\sim
2\Gamma\big{(}1+\frac{1}{r}\big{)}(er)^{1/r}n^{-1/r},\quad\text{as
}n\to\infty,$ (17)

which is known at least since the work [17] of Dirichlet.

###### Proof of Corollary 3.

We first prove the statement for $q<\infty$. For $X^{(n)}$ uniformly
distributed on $\mathbb{D}_{q,1}^{n}$, we can write

$\mathrm{vol}_{n}\big{(}\mathbb{D}_{q,1}^{n}\cap
t\mathbb{D}_{r}^{n}\big{)}=\mathbb{P}\Bigg{[}\|X^{(n)}\|_{r}\leq\frac{t}{\mathrm{vol}_{n}(\mathbb{B}_{r}^{n})^{1/n}}\Bigg{]}.$

Then, simply rewriting the previous expression, we obtain

$\mathrm{vol}_{n}\big{(}\mathbb{D}_{q,1}^{n}\cap
t\mathbb{D}_{r}^{n}\big{)}=\mathbb{P}\Bigg{[}n^{-1/r}\|X^{(n)}\|_{r}-m_{q,r}\leq\frac{t}{n^{1/r}\mathrm{vol}_{n}(\mathbb{B}_{r}^{n})^{1/n}}-m_{q,r}\Bigg{]},$

with $m_{q,r}$ as in Theorem D, in particular for $r<\infty$,

$m_{q,r}=\frac{1}{2e^{1/q}}\frac{q}{q-1}\left(\frac{\Gamma(r+1)\Gamma\Big{(}1+\frac{q}{q-1}\Big{)}}{\Gamma\Big{(}r+1+\frac{q}{q-1}\Big{)}}\right)^{1/r}.$

From the stated volume radius asymptotics in (17), we know that

$\frac{1}{n^{1/r}\mathrm{vol}_{n}(\mathbb{B}_{r}^{n})^{1/n}}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}c_{q,r}:=\begin{cases}\frac{1}{2(er)^{1/r}\Gamma(1+1/r)}&\colon
r<\infty\\\ \frac{1}{2}&\colon r=\infty.\end{cases}$

Therefore, in view of the weak law of large numbers in Theorem D, the
probability tends to one if $tc_{q,r}>m_{q,r}$ and to zero if
$tc_{q,r}<m_{q,r}$. Hence, the statement holds with threshold
$A_{q,r}=c_{q,r}(m_{q,r})^{-1}$.

If $q=\infty$, we can proceed analogously using
$m_{\infty,r}=\frac{1}{2}\Big{(}\frac{1}{r+1}\Big{)}^{1/r}$. This completes
the proof. ∎

##  4 The conjecture for general $p>1$ — Maximum entropy heuristics

Let us describe the heuristics leading us to Conjecture 1 and specifically to
the differential equation that appears in it. Our arguments are based on
maximum entropy considerations [52], but are not mathematically rigorous (see
[31] for a similar rigorous result in the case of Orlicz balls). We shall
denote by $\mathscr{M}_{1}(\mathbb{R}_{+})$ the set of probability measures on
$\mathbb{R}_{+}$ equipped with the weak topology, which is a Polish space.
Since Lorentz balls belong to the class of $1$-symmetric convex bodies, we can
restrict considerations to the positive orthant (see also Lemma 7) and
consider a uniformly distributed random vector $X^{(n)}$ in

$\widetilde{\mathbb{D}}_{q,p,+}^{n}=\Bigg{\\{}x\in\mathbb{R}^{n}_{+}\colon\frac{1}{n}\sum_{i=1}^{n}\Big{(}\frac{i}{n}\Big{)}^{p/q-1}|x_{i}^{*}|^{p}\leq
1\Bigg{\\}}.$

We now consider a sequence of independent and identically distributed “random
variables” $Y_{1},Y_{2},\dots$ which we assume to be uniformly “distributed”
according to the infinite Lebesgue measure $\lambda$ on $\mathbb{R}_{+}$.
Conditioned on
$Y^{(n)}:=(Y_{1},\dots,Y_{n})\in\widetilde{\mathbb{D}}_{q,p,+}^{n}$, which is
a so-called energy constraint, the “random vector” $Y^{(n)}$ is uniformly
distributed in $\widetilde{\mathbb{D}}_{q,p,+}^{n}$. The maximum entropy
principle states (under suitable conditions) that the random empirical
probability measure $L_{Y^{(n)}}$ associated to $Y^{(n)}$, where for
$x\in\mathbb{R}^{n}$

$L_{x}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}},$

converges, when conditioned upon the rare event of remaining in a closed
convex set $K\subset\mathscr{M}_{1}(\mathbb{R}_{+})$, to the measure $\mu^{*}$
minimizing the relative entropy

$H(\mu|\lambda):=\begin{cases}\int_{0}^{\infty}p(x)\log
p(x)\,\lambda(\mathrm{d}x)&:p=\frac{\mathrm{d}\mu}{\mathrm{d}\lambda}\text{
exists},\\\ \infty&\text{otherwise}\end{cases}$

over $K$ (c.f. Sanov’s theorem [52, Section 5.2] and its version for infinite
measures [6]). In order to rewrite the conditioning
$Y^{(n)}\in\widetilde{\mathbb{D}}_{q,p,+}^{n}$ in terms of the empirical
measure, we use the quantile function (i.e., the inverse cumulative
distribution function), which for any probability measure
$\mu\in\mathscr{M}_{1}(\mathbb{R}_{+})$ is given by

$Q_{\mu}\colon[0,1]\to[0,\infty],\quad Q_{\mu}(t):=\inf\big{\\{}s\geq
0\colon\mu([0,s])\geq t\big{\\}}.$

The following lemma follows from a simple rewriting which also appears in the
analysis of L-statistics, see, e.g., [11].

######  Lemma 8.

Let $n\in\mathbb{N}$. For any $x\in\mathbb{R}_{+}^{n}$, we have

$x\in\widetilde{\mathbb{D}}_{q,p,+}^{n}\,\,\Longleftrightarrow\,\,L_{x}\in
K_{n}:=\Bigg{\\{}\mu\in\mathscr{M}_{1}(\mathbb{R}_{+})\colon\int_{0}^{1}Q_{\mu}(t)^{p}J_{n}(t)\,\mathrm{d}t\leq
1\Bigg{\\}},$

where

$J_{n}(t):=\sum_{i=1}^{n}\Big{(}1-\frac{i-1}{n}\Big{)}^{p/q-1}\mathbbm{1}_{(\frac{i-1}{n},\frac{i}{n}]}(t).$

###### Proof.

By definition

$x\in\widetilde{\mathbb{D}}_{q,p,+}^{n}\quad\Longleftrightarrow\quad\frac{1}{n}\sum_{i=1}^{n}\Big{(}\frac{i}{n}\Big{)}^{p/q-1}|x_{i}^{*}|^{p}\leq
1.$

Moreover, we note that for $t\in[0,1]$

$Q_{L_{x}}(t)=\inf\big{\\{}s\geq 0\,:\,L_{x}([0,s])\geq
t\big{\\}}=\inf\Big{\\{}s\geq
0\,\colon\,\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}([0,s])\geq t\Big{\\}}\geq
0.$

The function $s\mapsto\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}([0,s])$ returns
the proportion of coordinates which are at most $s$. So the quantile function
evaluated at $t$ is the minimal $s$ such that this proportion still exceeds
$t$. If $t\in(\frac{i-1}{n},\frac{i}{n}]$, then this must be the value
$s=x_{n-i+1}^{*}$ (the $i^{\text{th}}$ smallest coordinate) since after this
point at least $i$ of the $n$ coordinates are at most $s$ and this is not true
for any smaller value. Therefore, we have

$Q_{L_{x}}(t)=x_{n-i+1}^{*}\quad\text{for
}t\in\Big{(}\frac{i-1}{n},\frac{i}{n}\Big{]},\,\,i\in\\{1,\dots,n\\}.$

Now we observe further that, for all $t\in[0,1]$,

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\Big{(}\frac{i}{n}\Big{)}^{p/q-1}|x_{i}^{*}|^{p}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\Big{(}1-\frac{i-1}{n}\Big{)}^{p/q-1}|x_{n-i+1}^{*}|^{p}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}J_{n}(t)Q_{L_{x}}(t)^{p}\mathbbm{1}_{(\frac{i-1}{n},\frac{i}{n}]}(t).$

Since both functions are constant on $t\in(\frac{i-1}{n},\frac{i}{n}]$, the
factor $1/n$ can be interpreted as the integral over these functions on this
interval, i.e.,

$\frac{1}{n}\sum_{i=1}^{n}J_{n}(t)Q_{L_{x}}(t)^{p}\mathbbm{1}_{(\frac{i-1}{n},\frac{i}{n}]}(t)=\sum_{i=1}^{n}\int_{\frac{i-1}{n}}^{\frac{i}{n}}J_{n}(t)Q_{L_{x}}(t)^{p}\,\mathrm{d}t=\int_{0}^{1}J_{n}(t)Q_{L_{x}}(t)^{p}\,\mathrm{d}t.$

This immediately proves the lemma. ∎

In view of Lemma 8, we have in our setting

$Y^{(n)}\in\widetilde{\mathbb{D}}_{q,p,+}^{n}\,\,\Longleftrightarrow\,\,L_{Y^{(n)}}\in
K_{n}.$

To apply variants of the Gibbs conditioning principle as the one used, e.g.,
in [21] and [36], we have to account for the fact that the set $K_{n}$ depends
on $n$. For large $n$ it will be approximately equal to the set

$K:=\Bigg{\\{}\mu\in\mathscr{M}_{1}(\mathbb{R}_{+})\colon\int_{0}^{1}Q_{\mu}(t)^{p}J(t)\,\mathrm{d}t\leq
1\Bigg{\\}},$

where

$J\colon[0,1]\to\mathbb{R},\quad J(t)=(1-t)^{p/q-1},\quad t\in[0,1],$

is the pointwise limit of the sequence $J_{n}$, $n\in\mathbb{N}$. We ignore
the technical details.

In order to minimize the relative entropy over all measures in $K$, we can
obviously ignore non-absolutely continuous measures for which the relative
entropy is $\infty$. Let $F_{\mu}$ be the distribution function of the
minimizer and $F_{\mu}^{\prime}=\frac{\mathrm{d}\mu}{\mathrm{d}\lambda}$ its
probability density function as well as $F_{\mu}^{-1}=Q_{\mu}$ the quantile
function. Using the substitution $x=F^{-1}_{\mu}(y)$ and the fact that
$Q_{\mu}^{\prime}(y)=(F_{\mu}^{-1})^{\prime}(y)=1/F_{\mu}^{\prime}(F_{\mu}^{-1}(y))$,
we can write

$H(\mu|\lambda)=\int_{0}^{\infty}F_{\mu}^{\prime}(x)\log
F_{\mu}^{\prime}(x)\,\lambda(\mathrm{d}x)=\int_{0}^{1}\log
F_{\mu}^{\prime}(F_{\mu}^{-1}(y))\,\lambda(\mathrm{d}y)=-\int_{0}^{1}\log
Q_{\mu}^{\prime}(y)\,\lambda(\mathrm{d}y).$

This leads to the following variational problem

$\displaystyle\max$ $\displaystyle\quad\int_{0}^{1}\log
Q^{\prime}(x)\,\mathrm{d}x$ s.t. $\displaystyle\quad
Q\colon[0,1]\to[0,\infty]$ $\displaystyle\quad Q\text{ is nondecreasing and
differentiable}$
$\displaystyle\quad\int_{0}^{1}Q(x)^{p}(1-x)^{p/q-1}\,\mathrm{d}x\leq 1.$

Its solution will be the quantile function of the minimizer $\mu^{*}\in K$ of
the relative entropy and thus of the supposed limiting distribution. For the
sake of convenience, we shall set $\alpha:=p/q-1\in[-1,0]$. In the following,
we heuristically compute a solution to this problem and derive a differential
equation for the corresponding distribution function.

Let us compute the variation and set $P:=Q+\varepsilon g$ for a small
$\varepsilon>0$ and a continuously differentiable function
$g\colon[0,1]\to\mathbb{R}$ such that $g(0)=0$:

$\int_{0}^{1}\log(Q+\varepsilon g)^{\prime}\mathrm{d}x=\int_{0}^{1}\log
Q^{\prime}(x)\,\mathrm{d}x+\int_{0}^{1}\log\Big{(}1+\varepsilon\frac{g^{\prime}(x)}{Q^{\prime}(x)}\Big{)}\,\mathrm{d}x.$

We expect that as $\varepsilon$ becomes small, the following approximation

$\int_{0}^{1}\log\Big{(}1+\varepsilon\frac{g^{\prime}(x)}{Q^{\prime}(x)}\Big{)}\,\mathrm{d}x=\varepsilon\int_{0}^{1}\frac{g^{\prime}(x)}{Q^{\prime}(x)}\,\mathrm{d}x+o(\varepsilon)$

is valid and hence

$\varepsilon^{-1}\Big{(}\int_{0}^{1}\log(Q+\varepsilon
g)^{\prime}\,\mathrm{d}x-\int_{0}^{1}\log
Q^{\prime}(x)\,\mathrm{d}x\Big{)}\stackrel{{\scriptstyle\varepsilon\to
0}}{{\longrightarrow}}\int_{0}^{1}\frac{g^{\prime}(x)}{Q^{\prime}(x)}\mathrm{d}x.$

We now look at the constraints and compute

$\int_{0}^{1}\big{(}Q(x)+\varepsilon
g(x)\big{)}^{p}(1-x)^{\alpha}\,\mathrm{d}x=\int_{0}^{1}Q(x)^{p}\Big{(}1+\varepsilon\frac{g(x)}{Q(x)}\Big{)}^{p}(1-x)^{\alpha}\,\mathrm{d}x.$

Again, we expect that, as $\varepsilon$ becomes small, the approximation

$\displaystyle\int_{0}^{1}Q(x)^{p}\Big{(}1+\varepsilon\frac{g(x)}{Q(x)}\Big{)}^{p}(1-x)^{\alpha}\,\mathrm{d}x$
$\displaystyle=\int_{0}^{1}Q(x)^{p}\Big{(}1+\varepsilon\frac{pg(x)}{Q(x)}\Big{)}(1-x)^{\alpha}\,\mathrm{d}x+o(\varepsilon)$
$\displaystyle=\int_{0}^{1}Q(x)^{p}(1-x)^{\alpha}\mathrm{d}x+\varepsilon\int_{0}^{1}pg(x)Q(x)^{p-1}(1-x)^{\alpha}\,\mathrm{d}x+o(\varepsilon)$

is valid. Hence,

$\varepsilon^{-1}\Big{(}\int_{0}^{1}(Q+\varepsilon
g)^{p}(1-x)^{\alpha}\mathrm{d}x-\int_{0}^{1}Q(x)^{p}(1-x)^{\alpha}\,\mathrm{d}x\Big{)}\stackrel{{\scriptstyle\varepsilon\to
0}}{{\longrightarrow}}\int_{0}^{1}pg(x)Q(x)^{p-1}(1-x)^{\alpha}\,\mathrm{d}x.$

Therefore, we only allow functions $g:[0,1]\to\mathbb{R}$ with $g(0)=0$ and

$p\int_{0}^{1}g(x)Q(x)^{p-1}(1-x)^{\alpha}\,\mathrm{d}x=0.$

A necessary condition for the maximizer is that for every such $g$,

$0=\int_{0}^{1}\frac{g^{\prime}(x)}{Q^{\prime}(x)}\,\mathrm{d}x=\frac{g(1)}{Q^{\prime}(1)}-\int_{0}^{1}g(x)\Big{(}\frac{1}{Q^{\prime}}\Big{)}^{\prime}(x)\,\mathrm{d}x=\int_{0}^{1}g(x)\Big{(}\frac{\delta_{1}(x)}{Q^{\prime}(1)}-\Big{(}\frac{1}{Q^{\prime}}\Big{)}^{\prime}(x)\Big{)}\,\mathrm{d}x,$

where in the second equality we used partial integration. We now use the
following result.

######  Lemma 9.

Let $\mu_{1},\mu_{2}$ be signed measures on $[0,1]$ such that if
$\int_{0}^{1}g(x)\mu_{1}(\mathrm{d}x)=0$ for some $g\in C[0,1]$ with $g(0)=0$,
then also $\int_{0}^{1}g(x)\mu_{2}(\mathrm{d}x)=0$. Then,
$\mu_{2}=\alpha\mu_{1}+\beta\delta_{0}$ for some constants
$\alpha,\beta\in\mathbb{R}$.

Applied to our situation above, we conclude that there exists some $c\neq 0$
such that, for all $x\in[0,1]$,

$\Big{(}\frac{1}{Q^{\prime}}\Big{)}^{\prime}(x)=cQ(x)^{p-1}(1-x)^{\alpha}\quad\text{and}\quad\frac{1}{Q^{\prime}(1)}=0.$

A substitution yields

$F^{\prime\prime}(x)=cF^{\prime}(x)(1-F(x))^{\alpha}x^{p-1},\quad
x\in\mathbb{R}_{+}.$

We must have $F(0)=0$ and there exists $r>0$ with $F(x)=1$ for $x\geq r$
($r=+\infty$ is possible). Also the condition on $Q$ translates to (if we
maximize)

$1=\int_{0}^{r}x^{p}(1-F(x))^{\alpha}F^{\prime}(x)\,\mathrm{d}x=\frac{1}{c}\int_{0}^{r}xF^{\prime\prime}(x)\,\mathrm{d}x.$

Using that $F^{\prime}(r)=0$, which follows from the fact
$Q^{\prime}(1)=+\infty$, together with integration by parts, we see that the
right-hand side is $-\frac{1}{c}$. Hence, we conclude $c=-1$ leading to the
desired differential equation.

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Zakhar Kabluchko: Faculty of Mathematics and Computer Science, University of
Münster, Orléans-Ring 10, 48149 Münster, Germany.

E-mail<EMAIL_ADDRESS>

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

E-mail<EMAIL_ADDRESS>

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

E-mail<EMAIL_ADDRESS>