# Probabilistic limit theorems induced by
the zeros of polynomials

Nils Heerten  Faculty of Mathematics, Ruhr University Bochum
<EMAIL_ADDRESS>,  Holger Sambale  Faculty of Mathematics,
Ruhr University Bochum<EMAIL_ADDRESS>and  Christoph Thäle
Department of Mathematics, Ruhr University Bochum  christoph.thaele@ruhr-uni-
bochum.de

###### Abstract.

Sequences of discrete random variables are studied whose probability
generating functions are zero-free in a sector of the complex plane around the
positive real axis. Sharp bounds on the cumulants of all orders are stated,
leading to Berry-Esseen bounds, moderate deviation results, concentration
inequalities and mod-Gaussian convergence. In addition, an alternate proof of
the cumulant bound with improved constants for a class of polynomials all of
whose roots lie on the unit circle is provided. A variety of examples is
discussed in detail.

###### Key words and phrases:

Berry-Esseen bound; combinatorial statistic; cumulant bound; Ehrhart
polynomial; Hurwitz polynomial; mod-Gaussian convergence; limit theorem;
moderate deviations; root unitary polynomial; zeros of polynomials

###### 2020 Mathematics Subject Classification:

05A16, 52B20, 60C05, 60F05, 60F15

##  1\. Introduction

Polynomials with non-negative coefficients are closely related to bounded
$\mathbb{N}_{0}$-valued random variables, where
$\mathbb{N}_{0}:=\\{0,1,2,\ldots\\}$, even by a one-to-one correspondence if
the sum of the coefficients is assumed to be equal to $1$. Indeed, the
probability generating function of any bounded $\mathbb{N}_{0}$-valued random
variable is a polynomial with non-negative coefficients which sum up to $1$,
and any such polynomial may be interpreted as the probability generating
function of a random variable taking finitely many values all of which are
non-negative integers.

It is therefore natural to search for connections between the location of the
zeros of the polynomials and asymptotic distributional properties of the
corresponding random variables. For instance, if the polynomials only have
real roots and thus can be written as the product of linear factors, the
respective random variables can be represented as sums of independent
Bernoulli variables in distribution. As a consequence, a central limit theorem
is satisfied if and only if the variance tends to infinity. For a survey of
this by now classical approach for proving probabilistic limit theorems, which
was originally introduced by L.H. Harper [15], we refer to the article of J.
Pitman [31].

For root-unitary polynomials, that is, polynomials all of whose roots lie on
the unit circle in the complex plane, H.-K. Hwang and V. Zacharovas [20]
proved (among other results) that in typical cases, the limiting behaviour of
sequences of the corresponding random variables $X_{N}$ is already determined
by the limit of the fourth moment of the standardizations
$X_{N}^{\ast}:=(X_{N}-\mathbb{E}X_{N})/\sqrt{\operatorname{var}(X_{N})}$. For
example, they proved the remarkable fact that $X_{N}^{\ast}$ converges to the
standard Gaussian random variable if and only if
$\mathbb{E}(X_{N}^{\ast})^{4}\to\leavevmode\nobreak\ 3$. This phenomenon can
be regarded as an instant of what is known as a fourth-moment phenomenon.

M. Michelen and J. Sahasrabudhe [27, 28] largely generalized this situation.
They proved central limit theorems and Berry-Esseen bounds on the speed of
convergence for sequences of bounded $\mathbb{N}_{0}$-valued random variables
$X_{N}$ in terms of the variance and a quantity $\delta_{N}$ which depends on
the “geometry” of the zero set of their generating functions $f_{N}$, which by
construction are polynomials. For instance, in [27, Theorem 1.1] $\delta_{N}$
equals $\min_{\zeta}|1-\zeta|$ with the minimum running over all roots of the
corresponding probability generating function $f_{N}$, while in Theorem 1.4
the polynomials are assumed to zero-free in a small sector
$\\{z\in\mathbb{C}\colon|\text{arg}(z)|<\delta_{N}\\}$, where
$\text{arg}(z)\in(-\pi,\pi]$ stands for the argument of $z\in\mathbb{C}$.

As a by-product and technical device for their main results, in the proof of
[27, Lemma 8.1] a bound on the cumulants of bounded $\mathbb{N}_{0}$-valued
random variables $X_{N}$ is derived which in particular applies to the
situation of [27, Theorem 1.4]. Writing
$\sigma_{N}^{2}:=\operatorname{var}(X_{N})$ and denoting the cumulant of order
$m$ of $X_{N}^{\ast}:=(X_{N}-\mathbb{E}X_{N})/\sigma_{N}$ by
$\kappa_{m}(X_{N}^{\ast})$, it says that

(1) $|{\kappa_{m}(X_{N}^{\ast})}|\leq{m!\over(c\delta_{N}\sigma_{N})^{m-2}}$

for some absolute constant $c>0$.

Cumulant bounds of this form contain a wealth of information on the asymptotic
distributional properties of the random variables $X_{N}$, which goes far
beyond of what has been studied in [27]. This approach was put forward by L.
Saulis and V. A. Statulevičius [34] (see also the recent survey [12]). In the
literature, a cumulant bound of the form

$|{\kappa_{m}(X_{N}^{\ast})}|\leq\frac{m!}{\Delta_{N}^{m-2}}\qquad(m\geq 3),$

is known as the _Statulevičius condition_ and gives rise to a large number of
strong limit theorems involving the quantity $\Delta_{N}$. In addition to
central limit theorems, this includes Berry-Esseen bounds, moderate deviation
principles and concentration inequalities. Moreover, the Statulevičius
condition can also be used to establish mod-Gaussian convergence, a fairly new
concept with far-reaching probabilistic consequences, which was introduced
only within the last decade. For details about the role of cumulant bounds in
mod-Gaussian convergence we refer to [13, Chapter 5]. In particular, mod-
Gaussian convergence leads to sharper versions of some of the bounds which can
be derived by the methods established in [34]. We remark that for the
coefficients of real-rooted polynomials, mod-Gaussian convergence is a
consequence of [13, Theorem 8.1].

In this note, we study cumulant bounds for sequences of bounded
$\mathbb{N}_{0}$-valued random variables and the distributional limit theorems
which follow from them together with a number of applications, mostly with a
combinatorial flavour. In particular, since (1) is slightly hidden in the
proofs of [27] and requires substantial theoretical and technical background,
we summarize the central arguments from [27] in a short and largely self-
contained proof. Another aim of this note is to provide an alternative and
much more elementary proof of (1) for a certain class of root-unitary
polynomials already appearing in [20], which at the same time leads to a far
better absolute constant $c$ in this special situation. It is based on
explicit representations of the cumulants $\kappa_{m}(X_{N}^{\ast})$ of any
order $m\geq 1$ for the random variables $X_{N}^{\ast}$ in [20].

The organization of this note is as follows. In Section 2 we state the
cumulant bound from [27] and various probabilistic limit theorems which follow
as corollaries. Their theoretical background is summarized in the appendix in
order to keep this paper largely self-contained. In Section 3 we consider a
class of root-unitary polynomials appearing in [20] and restate the cumulant
bound in this situation with improved constants. A number of examples are
discussed in Section 4. In the last section we provide the proofs of the
results from Section 2 (Section 5.1) and Section 3 (Sections 5.2 and 5.3).

##  2\. Main results

Throughout this note, we shall consider sequences of bounded
$\mathbb{N}_{0}$-valued random variables $(X_{N})_{N}$, i.e.,
$X_{N}\in\\{0,1,\ldots,n(N)\\}$ for a suitable natural number $n(N)$. We
denote their probability generating functions (which are polynomials by
assumption) by $f_{N}:=f_{X_{N}}$ and their variances by $\sigma^{2}_{N}>0$.
We further assume the generating functions $f_{N}$ to be zero-free in the
sector

$S(\delta_{N}):=\big{\\{}z\in\mathbb{C}\setminus\\{0\\}:\arg(z)\in(-\delta_{N},\delta_{N})\big{\\}}\qquad(\delta_{N}\in(0,\pi)),$

$1$$\mathfrak{Re}$$\mathfrak{Im}$$\delta_{N}$$0$ Figure 1. The sector
$S(\delta_{N})$ (blue) in the complex plane.

adopting the notation used in [27]. See Figure 1 for a visualization.

We denote the cumulant of order $m$ of the random variable $X_{N}$ by
$\kappa_{m,N}:=\kappa_{m}(X_{N})$. The normalization of the random variables
will be short-handed by $X_{N}^{\ast}:=(X_{N}-\mathbb{E}X_{N})/\sigma_{N}$ and
the respective cumulants of order $m$ of $X_{N}^{\ast}$ will be denoted by
$\kappa_{m,N}^{\ast}:=\kappa_{m}(X_{N}^{\ast})$. Throughout this note, $c$
will always indicate an absolute constant whose value might differ from
occasion to occasion.

The first theorem provides a bound on the cumulants of random variables whose
generating functions have no zeros in $S(\delta_{N})$. As mentioned in the
introduction, this result is taken from the proofs given in [27] (see
especially Lemma 8.1 therein). However, due to the variety of implications it
draws, we highlight it at this point and recall the arguments of [27] in a
short and mostly self-contained proof in Section 5.1.

######  Theorem 2.1.

Let $(X_{N})_{N\geq 1}$ be a sequence of bounded random variables taking
values in $\mathbb{N}_{0}$. For $\delta_{N}\in(0,\pi)$, assume the probability
generating functions $f_{N}$ to have no roots lying in the sector
$S(\delta_{N})$. Then, for $m\geq 3$,

(2) $|{\kappa_{m,N}^{\ast}}|\leq\frac{m!}{(c\delta_{N}\sigma_{N})^{m-2}},$

where $c>0$ is some absolute constant.

As we will confirm in the proof, the arguments in [27] lead to $c=2^{-3248}$
for the constant in Theorem 2.1.

The Statulevičius condition (2) opens the door to a variety of remarkable
probabilistic implications, of which we will discuss a few exemplarily. We
briefly summarize these in Lemma A.1 in the appendix and refer to the
monograph [34] as well as the survey article [12] for further details. The
first of them provides a Berry-Esseen type bound on the Kolmogorov distance of
$X_{N}^{\ast}$ and a standard Gaussian random variable.

######  Corollary 2.2.

Let $(X_{N})_{N\geq 1}$ be a sequence of random variables as in Theorem 2.1.
Then, for a standard Gaussian random variable $Z$,

$\sup_{x\in\mathbb{R}}\big{|}{\mathbb{P}}(X_{N}^{\ast}\leq
x)-{\mathbb{P}}(Z\leq x)\big{|}\leq\frac{c}{\delta_{N}\sigma_{N}}\qquad(N\geq
1),$

where $c>0$ is some absolute constant. In particular, if
$\delta_{N}\sigma_{N}\to\infty$ as $N\to\infty$, $X_{N}^{\ast}$ converges to
$Z$ in distribution.

Combining the constant from Theorem 2.1 with Lemma A.1 (i) (for $\gamma=0$),
the constant in Corollary 2.2 can be chosen as $c=324\sqrt{2}\cdot
2^{3248}=81\sqrt{2}\cdot 2^{3250}$.

The cumulant bound (2) also leads to moderate deviation results. In the
sequel, we write $a_{N}=o(b_{N})$ for two sequences of real numbers
$(a_{N})_{N\geq 1}$ and $(b_{N})_{N\geq 1}$ such that $a_{N}/b_{N}\to 0$ as
$N\to\infty$.

######  Corollary 2.3.

Let $(X_{N})_{N\geq 1}$ be a sequence of random variables as in Theorem 2.1.
Further, let $(a_{N})_{N\geq 1}$ be a sequence of real numbers with
$a_{N}\to\infty$ as $N\to\infty$ and $a_{N}=o(\delta_{N}\sigma_{N})$. Then,

$\lim_{N\to\infty}\frac{1}{a_{N}^{2}}\log{\mathbb{P}}\big{(}X_{N}^{\ast}/a_{N}\in(x,\infty)\big{)}=-\frac{x^{2}}{2}\qquad(x\geq
0).$

Corollary 2.3 can be understood as a special case of a moderate derivation
principle. In fact, under the same assumptions, a full moderate deviation
principle (MDP) with speed $a_{N}^{2}$ and rate function $I(x)=x^{2}/2$ for
the sequence $(X_{N}^{\ast}/a_{N})_{N\geq 1}$ is valid, see [12, Theorem 3.1].
In our case, the sequence $(X_{N}^{\ast}/a_{N})_{N\geq 1}$ satisfies an MDP
with

$\displaystyle\liminf_{N\to\infty}\frac{1}{a_{N}^{2}}\log{\mathbb{P}}\big{(}X_{N}^{\ast}/a_{N}\in
A\big{)}$ $\displaystyle\geq-\inf_{x\in A^{\circ}}\frac{x^{2}}{2},$
$\displaystyle\limsup_{N\to\infty}\frac{1}{a_{N}^{2}}\log{\mathbb{P}}\big{(}X_{N}^{\ast}/a_{N}\in
A\big{)}$ $\displaystyle\leq-\inf_{x\in\bar{A}}\frac{x^{2}}{2},$

for any Borel set $A\subset\mathbb{R}$, where $A^{\circ}$ and $\bar{A}$ denote
the interior and the closure of $A$, respectively. This includes Corollary 2.3
as a special case.

Theorem 2.1 furthermore implies mod-Gaussian convergence for the sequence of
random variables $X_{N}$, which is a special case of mod-$\phi$ convergence.
We recall from [13, 26] that a sequence of random variables $X_{N}$ converges
in the mod-Gaussian sense with parameters $t_{N}\to\infty$ and limiting
function $\Psi$, provided that

$\lim_{N\to\infty}\mathbb{E}[e^{zX_{N}}]e^{-t_{N}{z^{2}\over 2}}=\Psi(z)$

locally uniformly on $\mathbb{C}$, where $\Psi$ is a non-degenerate analytic
function.

######  Corollary 2.4.

Let $(X_{N})_{N\geq 1}$ be a sequence of random variables as in Theorem 2.1,
and let $\Delta_{N}:=c\delta_{N}\sigma_{N}$, where $c$ is the absolute
constant from (2).

  * (i)

Assuming that $\kappa_{3,N}^{\ast}\Delta_{N}\to L\in\mathbb{R}$ as
$N\to\infty$, the sequence of random variables $\Delta_{N}^{1/3}X_{N}^{\ast}$
converges in the mod-Gaussian sense with parameters $\Delta_{N}^{2/3}$ and
limiting function $e^{{L\over 6}z^{3}}$.

  * (ii)

Assuming that $\kappa_{3,N}^{\ast}=0$ and
$\kappa_{4,N}^{\ast}\Delta_{N}^{2}\to L\in\mathbb{R}$ as $N\to\infty$, the
sequence of random variables $\sqrt{\Delta_{N}}X_{N}^{\ast}$ converges in the
mod-Gaussian sense with parameters $\Delta_{N}$ and limiting function
$e^{{L\over 24}z^{4}}$.

Natural situations in which part (ii) applies include all polynomials
discussed in Section 3. We remark that mod-Gaussian convergence gives rise to
various further implications as Cramér-Petrov type large deviations and
precise deviations (see [13, Chapter 5.2]). The latter means an estimate for
probabilities of the type
${\mathbb{P}}\big{(}X_{N}^{\ast}\in(x,\infty)\big{)}$ on a non-logarithmic
scale for suitable values of $x$, which are allowed to depend on the parameter
$N$ and which involve the limiting function $\Psi$. Since such results are
rather technical to formulate, we refrain from presenting them. Instead, we
mention the following concentration inequality, which yields a Bernstein-type
bound for the upper tails of $X_{N}^{\ast}$.

######  Corollary 2.5.

Let $(X_{N})_{N\geq 1}$ be a sequence of random variables as in Theorem 2.1,
and let $\Delta_{N}:=c\delta_{N}\sigma_{N}$, where $c$ is the absolute
constant from (2). Then, there exists an absolute constant $C>0$ such that for
all $x\geq 0$,

${\mathbb{P}}(X_{N}^{\ast}\geq x)\leq
C\exp\Big{(}-\frac{1}{2}\frac{x^{2}}{2+x/\Delta_{N}}\Big{)}.$

##  3\. Results for a class of root-unitary polynomials

All over this section, we consider $\mathbb{N}_{0}$-valued random variables
$X_{N}$ whose probability generating functions admit a representation of the
form

(3) $f_{N}(z)=f_{X_{N}}(z):=\frac{P_{N}(z)}{P_{N}(1)}\qquad(z\in\mathbb{R}),$

where

(4) $P_{N}(z):=\prod_{1\leq j\leq N}\frac{1-z^{b_{j}}}{1-z^{a_{j}}},$

for some natural number $N$ and exponents $a_{j},b_{j}\in\mathbb{N}$,
$j=1,\dots,N$. We assume that $b_{j}\geq a_{j}$ for all $j=1,\dots,N$ and note
that $P_{N}(1)=\prod_{1\leq j\leq N}b_{j}/a_{j}$, which follows readily from
the fact that $(1-z^{k})=(1-z)(1+z+\ldots+z^{k-1})$ for $k\geq 1$. The degree
of $P_{N}$ is denoted by $n$ and is related to the parameter $N$ by
$n=\sum_{1\leq j\leq N}(b_{j}-a_{j})$. As opposed to the probability
generating function (3), we refer to $P_{N}$ as the generating polynomial of
$X_{N}$ for the remainder of this section.

The probability generating functions $f_{N}$ defined by (3) have roots all
located on the unit circle in the complex plane, i.e., they are _root-unitary_
in the terminology of [20]. More precisely, their roots are a subset of
$\\{\exp(\pm 2\pi ik/b_{j}),k=1,\ldots,\lfloor
b_{j}/2\rfloor,j=1,\ldots,N\\}$, $1$ is never a root (since $P_{N}(1)\neq 0$)
and we may assume that $\max_{j}a_{j}<\max_{j}b_{j}$. In particular, the
functions $f_{N}$ have no zeros in $S(\delta_{N})$ for
$\delta_{N}:=2\pi/\max_{j=1,\ldots,N}b_{j}$.

Probability generating functions of type (3) have been studied in [20, Section
4], where they form an important subclass of root-unitary polynomials, and
previously also in [10, Section 3] in the more specific context of $q$-Catalan
numbers. We emphasize that we use a different notation as in [20], who index
(3) and (4) by the degree $n$. Random variables with probability generating
functions of type (3) naturally appear in a variety of examples, a few of
which will be discussed in Section 4.3.

A reformulation of Theorem 2.1 for the probability generating functions (3)
admits a more straightforward proof by elementary means, which we present in
Section 5.3. Its core ingredient is an explicit representation of the
cumulants of the corresponding random variables $X_{N}$. For $m\geq 3$, they
are given by

(5) $\kappa_{m,N}=\frac{B_{m}}{m}\sum_{1\leq j\leq N}(b_{j}^{m}-a_{j}^{m}),$

where $B_{m}$ denotes the $m$-th Bernoulli number, see Section 5.2 for a
proof. In particular, since $B_{2m+1}=0$ for any $m\geq 1$, all cumulants of
odd order $\geq 3$ vanish identically, and when establishing cumulant bounds
it is therefore sufficient to consider the cumulants of even orders $2m$.

######  Theorem 3.1.

Let $P_{N}(z)$ be a polynomial as in (4) and $X_{N}$ a random variable defined
as in (3). Then, for $m\geq 2$, the $2m$-th cumulant of $X_{N}^{\ast}$
satisfies

$|\kappa_{2m,N}^{\ast}|\leq\frac{(2m)!}{\Delta_{N}^{2m-2}}$

with

$\Delta_{N}=\pi^{2}\sqrt{\frac{7}{6}}\frac{\sigma_{N}}{M_{N}},$

where $M_{N}=\max_{1\leq j\leq N}b_{j}$.

As noted above, we may set $\delta_{N}=2\pi/M_{N}$ in the notation of Section
2, so that the bound given in Theorem 3.1 reads

$|\kappa_{2m,N}^{\ast}|\leq\frac{(2m)!}{(c\delta_{N}\sigma_{N})^{2m-2}}$

with $c=\pi\sqrt{7/24}$. In particular, this gives back (2) but with a
remarkably better absolute constant (which also extends to the corollaries
stated in Section 2, so that we may choose $c=324\pi\sqrt{7/12}$ in Corollary
2.2, for instance).

Note that [18, Lemma 4] also yields Berry-Esseen type bounds for sequences of
random variables whose probability generating functions factorize into
polynomials with non-negative coefficients. In particular, assuming all the
factors $(1-z^{b_{j}})/(1-z^{a_{j}})$ in (4) to be polynomials with non-
negative coefficients, by [18, Lemma 4] a Berry-Esseen bound

$\sup_{x\in\mathbb{R}}\big{|}{\mathbb{P}}(X_{N}^{\ast}\leq
x)-{\mathbb{P}}(Z\leq x)\big{|}\leq\frac{c}{\Delta_{N}^{\prime}}$

holds with $\Delta_{N}=\sigma_{N}/M_{N}^{\prime}$, where
$M_{N}^{\prime}:=\max_{1\leq j\leq N}(b_{j}-a_{j})$. Note that in all examples
discussed in Section 4.3, the asymptotic behaviour of $M_{N}$ and
$M_{N}^{\prime}$ (whenever applicable) coincides. On the other hand, some of
the examples we treat do not fall into the class of random variables
considered in [18].

##  4\. Examples

###  4.1. Polynomials with roots only on the negative half-axis

One of the simplest situations in which our results apply are polynomials
which have only real roots, all of which are negative (so that they are zero-
free in $S(\delta)$ for any $\delta\in(0,\pi]$). As already mentioned in the
introduction, central limit theorems for such situations are rather classical
and go back to the work of L.H. Harper [15] and many examples can be found,
for example in [31]. We shall exemplarily discuss one further application,
which deals with descents in finite Coxeter groups, a topic we will pick up
below once again.

Given a permutation $\pi$ on the set $\\{1,\dots,N\\}$, the descents of $\pi$
are defined as

$\text{Des}(\pi)=\big{\\{}i:1\leq i<N,\pi(i)>\pi(i+1)\big{\\}}.$

Descents of permutations can be regarded as a special type of the more general
theory of descents in finite Coxeter groups. For a detailed introduction to
Coxeter groups we refer to [8, Chapter 1]. For simplicity, we only consider
the following three classical types.

The Coxeter group of type $A_{N}$ corresponds to the permutations on
$\\{1,\ldots,N+1\\}$. The Coxeter group of type $B_{N}$ can be realized as the
group of signed permutations, that is, the group of all bijections $\pi$ on
$\\{\pm 1,\dots,\pm N\\}$, such that $\pi(-i)=-\pi(i)$, see [8, Chapter 8.1].
Following the one-line notation of [22, Section 2], we write
$\pi=[\pi(1),\dots,\pi(N)]$, with $\pi(i)\in\\{\pm 1,\dots,\pm N\\}$ and
$\\{|\pi(1)|,\dots,|\pi(N)|\\}=\\{1,\dots,N\\}$. The Coxeter group of type
$D_{N}$ can then be realized as a subgroup of $B_{N}$ with an even number of
negative entries in the one-line notation, in other words,

$D_{N}=\big{\\{}\pi\in B_{N}\>:\>\pi(1)\pi(2)\cdots\pi(N)>0\big{\\}},$

see [8, Chapter 8.2]. Note that the groups $A_{N},B_{N}$ and $D_{N}$ all have
rank $N$.

In accordance with Proposition 8.1.2 and Proposition 8.2.2 in [8] we set

$\pi(0):=\begin{cases}0&\text{if }\pi\in A_{N-1}\text{ or }B_{N},\\\
-\pi(2)&\text{if }\pi\in D_{N}.\end{cases}$

The descents of an element $\pi\in A_{N-1},B_{N},D_{N}$ can now be defined as

$\text{Des}(\pi):=\big{\\{}i:0\leq i<N,\pi(i)>\pi(i+1)\big{\\}},$

see [22, Section 2]. We refer to [25] for a discussion of the more general
concept of $d$-descents in the classical types of Coxeter groups.

If we set $W_{N}$, $W\in\\{A,B,D\\}$, to be a finite Coxeter group of rank
$N$, the number of elements in $W_{N}$ with exactly $k\in\\{0,1,\dots,N\\}$
descents is called the $W_{N}$-Eulerian number, see [22]. If we define a
random variable $X_{N}$ as the number of descents of an element of $W_{N}$
chosen uniformly at random, the distribution of $X_{N}$ is called the
$W_{N}$-Eulerian distribution. The generating function of $X_{N}$ is given by

$f_{N}(z)=\frac{P_{N}(z)}{P_{N}(1)},\qquad\text{where
}P_{N}(z)=\prod_{i=1}^{N}(z+q_{i})$

for some positive real numbers $q_{1},\dots,q_{N}$ (see [22, Theorem 2.4]) and
hence only has real-valued, negative roots. Therefore, Theorem 2.1 can be
applied with any choice of $\delta_{N}\in(0,\pi]$.

Since $\sigma_{N}^{2}\sim N/12$ by [22, Corollary 4.2], we obtain from
Corollary 2.2 a central limit theorem with speed of convergence of order
$\mathcal{O}(N^{-1/2})$. This is consistent with the results of [22, Theorem
6.2] and [25, Corollary 2.7]. According to [21, Theorem 5.3], in the case
$W_{N}=A_{N}$, the cumulants of order $m$ are given by

$\kappa_{m,N}=(N+1){B_{m}\over m}\qquad(2\leq m\leq N),$

where $B_{m}$ denotes the $m$-th Bernoulli number. Therefore, all odd
cumulants vanish, since $B_{2m+1}=0$. In particular,
$\kappa_{4,N}=-(N+1)/120$, $\kappa_{4,N}\Delta_{N}^{2}\to-\pi^{2}c^{2}/10$, as
$N\to\infty$, and hence Corollary 2.4 implies convergence of
$\sqrt{\Delta_{N}}X_{N}^{\ast}$ in the mod-Gaussian sense with parameter
$\Delta_{N}$ and limiting function $\exp(-c^{2}\pi^{2}z^{4}/240)$, for $X_{N}$
being the number of descents in an element of the Coxeter group $A_{N}$ and
with $c$ being the absolute constant from (2). The cases $W_{N}=B_{N}$ and
$W_{N}=D_{N}$ can be handled similarly.

######  Remark 4.1.

As mentioned in [22, Theorem 2.4], the representation of the generating
function $f_{N}(z)$ given above holds for descents in finite Coxeter groups in
general and hence all consequences remain valid in such a more general set-up.
We chose to consider Coxeter groups of type $A_{N}$, $B_{N}$ and $D_{N}$ only
since a representation of descents in general finite Coxeter groups would
require the study of root systems of these groups and thus significantly
complicates the presentation. We refer to [22] and [25] for a detailed
discussion.

Figure 2. Location of the roots of the generating polynomials of the
conditional binomial distribution (left panel) with $N=25$, $p=1/2$ (blue),
$p=1/3$ (yellow), $p=1/4$ (green), $p=2/3$ (orange) and $p=3/4$ (violet) and
of the alternating descents (right panel) with $N=25$ in the complex plane.

###  4.2. Hurwitz polynomials

A Hurwitz (or Hurwitz-stable) polynomial is a polynomial all of whose roots
$\zeta$ lie in the open left half-plane, i.e., $\mathfrak{Re}(\zeta)<0$. Such
a polynomial is again zero-free in $S(\delta)$ for any $\delta\in(0,\pi/2)$.
In particular, the polynomials discussed in Section 4.1 form a subclass of
Hurwitz polynomials. In the sequel, we discuss several examples of Hurwitz
polynomials with non-real roots.

####  4.2.1. Conditional binomial distribution

We say that a random variable $X_{N}$, $N\in\mathbb{N}$, has conditional
binomial distribution $\operatorname{CoBin}(N,p)$ with parameters $N$ and
$p\in(0,1)$, if for all $k=0,\ldots,N-1$,

${\mathbb{P}}(X_{N}=k):={\mathbb{P}}(Y_{N}=k\mid Y_{N}\leq N-1),$

where $Y_{N}\sim\operatorname{Bin}(N,p)$ has binomial distribution with
parameter $p$. This slightly generalizes the random variables discussed in
Lemma 4 and Theorem 3 in [6] (which correspond to the case of $p=1/2$). For
each $N$, $X_{N}$ has probability generating function

$f_{N}(z)={1\over
1-p^{N}}\sum_{k=0}^{N-1}z^{k}\binom{N}{k}p^{k}(1-p)^{N-k}={\big{(}p(z-1)+1\big{)}^{N}-(zp)^{N}\over
1-p^{N}}.$

To see that $f_{N}(z)$ is a Hurwitz polynomial, note that the equation
$f_{N}(z)=0$ is equivalent to

$\displaystyle 0$ $\displaystyle=\big{(}zp+(1-p)\big{)}^{N}-(zp)^{N},$

which leads to solving

$\Big{(}1+\frac{1-p}{zp}\Big{)}^{N}=1.$

Consequently, we obtain

$1+\frac{1-p}{z_{k}p}=\cos\Big{(}\frac{2\pi
k}{N}\Big{)}+i\sin\Big{(}\frac{2\pi k}{N}\Big{)},\quad k=0,\ldots,N-1,$

which yields

$z_{k}=\frac{1}{2}\Big{(}\frac{1}{p}-1\Big{)}\Big{(}-1-i\cot\Big{(}\frac{\pi
k}{N}\Big{)}\Big{)},\qquad k=1,\dots,N-1,$

and therefore all roots $\zeta$ of $f_{N}(z)$ satisfy
$\mathfrak{Re}(\zeta)=(1-1/p)/2<0$. The location of these roots for various
$p$ is shown in the left panel of Figure 2. As a consequence, Theorem 2.1
applies for any choice of $\delta_{N}\in(0,\pi/2)$, leading to a bound on the
cumulants of $X_{N}^{\ast}$. It then is straightforward to show that

$\displaystyle\mathbb{E}X_{N}=N\frac{p-p^{N}}{1-p^{N}}\qquad\text{and}\qquad\sigma_{N}^{2}=N\frac{p(1-p)}{1-p^{N}}-N^{2}\frac{p^{N}((p-1)^{2})}{(1-p^{N})^{2}}.$

Note that for fixed $p\in(0,1)$, we have $\sigma_{N}^{2}=Np(1-p)+o(1)$ such
that $\sigma_{N}^{2}\to\infty$ as $N\to\infty$ and hence all the asymptotic
distributional results from Section 2 apply. In particular, we obtain a
central limit theorem with speed of convergence of order
$\mathcal{O}(N^{-1/2})$ given by Corollary 2.2. Moreover,

$\kappa_{3,N}=Np(1-p)(1-2p)+o(1),$

which yields
$\kappa_{3,N}^{\ast}\sigma_{N}=\kappa_{3,N}\sigma_{N}^{-2}\rightarrow 1-2p$ as
$N\to\infty$. This implies convergence of $\Delta_{N}^{1/3}X_{N}^{\ast}$ in
the mod-Gaussian sense according to Corollary 2.4 with parameter
$\Delta_{N}^{2/3}$ and limiting function $\exp((1-2p)c\pi z^{3}/12)$, where
$c$ is the absolute constant from (2).

####  4.2.2. Ehrhart polynomials

Taking $p=1/2$ in the previous example, we saw that all roots of the
probability generating function $f_{N}$ of $X_{N}$ were located on the line
$\\{z\in\mathbb{C}:\mathfrak{Re}(z)=-1/2\\}$. A similar property has become
rather prominent in the theory of lattice polytopes of which we can outline
only selected aspects and refer to [5] for further details. We recall that a
polytope $Q\subset\mathbb{R}^{N}$ is called a lattice polytope if all of its
vertices have integer coordinates. A lattice polytope $Q$ is reflexive if also
its convex dual $Q^{\circ}:=\\{x\in\mathbb{R}^{N}:\langle x,y\rangle\leq 1\
\text{for all}\ y\in Q\\}$ is a lattice polytope, where
$\langle\,\cdot\,,\,\cdot\,\rangle$ denotes the standard scalar product in
$\mathbb{R}^{N}$. It is well known and the content of Ehrhart’s theorem [5,
Theorem 3.8] that for a lattice polytope $Q$ the counting function

$E_{Q}(k):=|\\{t\in\mathbb{Z}^{N}:t\in kQ\\}|,\qquad k\in\mathbb{N}_{0},$

is the evaluation of a polynomial $E_{Q}(z)$, $z\in\mathbb{R}$, of degree $N$,
the so-called Ehrhart polynomial of $Q$. To interpret such a polynomial as the
probability generating polynomial of a random variable, we will assume from
now on that $Q$ is Ehrhart positive, meaning that all coefficients of
$E_{Q}(z)$ are positive. Finally, a lattice polytope $Q$ is called a CL-
polytope, provided that all roots of its Ehrhart polynomial are located on the
so-called canonical line $\\{z\in\mathbb{C}:\mathfrak{Re}(z)=-1/2\\}$.

For example, if $W_{N}:=[0,1]^{N}$ is the $N$-dimensional unit cube, then

$E_{W_{N}}(z)=(z+1)^{N},$

see [5, Theorem 2.1]. From this we conclude that the random variable with
probability generating function $E_{W_{N}}(z)/E_{W_{N}}(1)=(z+1)^{N}/2^{N}$
follows a binomial distribution with parameter $N$ and $1/2$. However, since
$z=-1$ is the only root of $E_{W_{N}}$, $W_{N}$ is not a CL-polytope, but it
corresponds to a real-rooted polynomial as discussed in Section 4.1.

Other examples connected to the binomial distribution are root polytopes of
type $A$ and $C$ and their convex duals. If $e_{1},\ldots,e_{N}$ denotes the
standard orthonormal basis in $\mathbb{R}^{N}$ the root polytopes of type $A$
and $C$ are defined, respectively, as the convex hulls of the classical root
systems of type $A$ and $C$, that is,

$\displaystyle A_{N}$
$\displaystyle:=\mathrm{conv}(\\{\pm(e_{i}+\ldots+e_{j}):1\leq i\leq j\leq
N\\}),$ $\displaystyle C_{N}$
$\displaystyle:=\mathrm{conv}(\\{\pm(2e_{i}+\ldots+2e_{N-1}+e_{N}):1\leq i\leq
N-1\\}\cup\\{\pm e_{N}\\}).$

The Ehrhart functions of both $A_{N}$ and $C_{N}$ are known explicitly,
showing in particular that both polytopes are of class CL, see [16, Sections 3
and 5] . For the dual convex polytopes $A_{N}^{\circ}$ and $C_{N}^{\circ}$
from [16, Lemma 5.3] and [17, Theorem 1.1] one has that

$E_{A_{N}^{\circ}}(z)=(z+1)^{N+1}-z^{N+1}\qquad\text{and}\qquad
E_{C_{N}^{\circ}}(z)=(z+1)^{N}+z^{N},$

implying that also $A_{N}^{\circ}$ and $C_{N}^{\circ}$ are CL-polytopes.
Moreover, up to the normalization constant $E_{A_{N-1}^{\circ}}(1)=2^{N}-1$,
$E_{A_{N-1}^{\circ}}(z)$ coincide with the probability generating function
$f_{N}$ for $p=1/2$ considered in the previous section. In other words, the
random variable connected to $A_{N}^{\circ}$ has the conditional binomial
distribution $\mathrm{CoBin}(N,1/2)$. For a probabilistic interpretation of
$E_{C_{N}^{\circ}}(z)$, let $U_{N}$ and $V_{N}$ be independent random
variables, $U_{N}$ having a binomial distribution with parameters $N$ and
$1/2$, while $V_{N}$ has a Bernoulli distribution with success probability
$2^{N}/(2^{N}+1)$. Consider the random variable $N-U_{N}V_{N}$. By
conditioning on the value of $V_{N}$, one easily check that its probability
generating function coincides with
$E_{C_{N}^{\circ}}(z)/E_{C_{N}^{\circ}}(1)$.

The Ehrhart polynomials of CL-polytopes are thus Hurwitz polynomials and the
random variables generated by the coefficients of the normalizations of these
polynomials satisfy the asymptotic distributional results derived in Section
2.

####  4.2.3. Alternating descents in permutations

Recalling the definition of descents of a permutation $\pi$ on the set
$\\{1,\dots,N\\}$ from Section 4.1, the alternating descents of $\pi$ are the
positions $i$ such that $\pi(i)>\pi(i+1)$ if $i$ is odd, or $\pi(i)<\pi(i+1)$
if $i$ is even. That is, an alternating descent is a descent if $i$ is odd and
an ascent if $i$ is even. It was shown in [9] that the number of alternating
descents agrees with the number of $3$-descents of permutations on
$\\{1,\ldots,N+1\\}$ with $\pi(1)=1$. Here, a $3$-descent is an index $i$ such
that $\pi(i)\pi(i+1)\pi(i+2)$ forms an odd permutation of size $3$, i.e., it
has one of the patterns $132$, $213$, or $321$.

Let $X_{N}$ be the number of alternating descents of a permutation $\pi$ on
$\\{1,\ldots,N\\}$ chosen uniformly at random. In [23, Theorem 4], it was
shown that all the roots $\zeta$ of the generating function $f_{N}(z)$ of
$X_{N}$ satisfy $|{\zeta}|=1$ and have a strictly negative real part. That is,
they all lie on the left half on the unit circle in the complex plane, see the
right panel of Figure 2. In other words, $f_{N}(z)$ is a Hurwitz polynomial,
so that all the results from Section 2 apply for any choice of
$\delta_{N}\in(0,\pi/2)$. It holds that $\sigma_{N}^{2}=(5N+3)/12$, so that
Corollary 2.2 provides a central limit theorem with Berry-Esseen bound of
order $\mathcal{O}(N^{-1/2})$, which confirms the speed of convergence
established in [19, Section 5.4.2]. Our results complement these findings by
moderate deviation and mod-Gaussian convergence results. For the latter, note
that $\kappa_{3,N}=0$ and $\kappa_{4,N}=-(81N-78)/120$ for $N\geq 3$ as can be
computed from the explicit exponential generating function reported in [23,
Equation (1)] or [19, Section 5.4.2]. This, together with the choice of
$\delta_{N}\equiv\pi/2$, leads to
$\kappa_{4,N}^{\ast}\Delta_{N}^{2}\to-\frac{81c^{2}\pi^{2}}{200}$ as
$N\to\infty$. Hence, by Corollary 2.4, $\sqrt{\Delta_{N}}X_{N}^{\ast}$, where
$c$ is the absolute constant from (2), converges in the mod-Gaussian sense
with parameter $\Delta_{N}=\pi(5N+3)c/24$ and limit function
$\exp(-27\pi^{2}c^{2}z^{4}/1600)$.

###  4.3. Root unitary polynomials

In this section, we discuss various examples of polynomials which are zero-
free in some sector $S(\delta_{N})$ with $\delta_{N}\to 0$ as $N\to\infty$. In
particular, these examples provide natural appearances of generating functions
that are root-unitary polynomials as studied in Section 3.

####  4.3.1. Inversions in Coxeter groups

Figure 3. Location of the roots of the generating polynomial of the number of
inversions in a Coxeter group of type $A_{N}$ with $N=10$ (left panel) and
$N=25$ (right panel) in the complex plane.

An inversion of a permutation $\pi$ on the set $\\{1,\dots,N\\}$ is a pair
$(i,j)$ with $1\leq i<j\leq N$ and $\pi(i)>\pi(j)$. Similar to descents in
Section 4.1, inversions of permutations can be regarded as a special type of
the more general theory of inversions in finite Coxeter groups, where as above
we only consider the three classical types $A_{N},B_{N}$ and $D_{N}$. For any
of these three groups, certain elements may be identified as inversions.
Indeed, according to the notation in [22, Section 2], we set

$\displaystyle\operatorname{inv}^{+}(\pi)$ $\displaystyle:=\\{1\leq i<j\leq
N:\pi(i)>\pi(j)\\},$ $\displaystyle\operatorname{inv}^{-}(\pi)$
$\displaystyle:=\\{1\leq i<j\leq N:-\pi(i)>\pi(j)\\},$
$\displaystyle\operatorname{inv}^{\circ}(\pi)$ $\displaystyle:=\\{1\leq i\leq
N:\pi(i)<0\\}.$ Then, the inversions in an element $\pi\in W_{N}$,
$W\in\\{A,B,D\\}$ are defined as
$\displaystyle\operatorname{inv_{A_{N-1}}}(\pi)$
$\displaystyle:=\operatorname{inv}^{+}(\pi),$
$\displaystyle\operatorname{inv_{B_{N}}}(\pi)$
$\displaystyle:=\operatorname{inv}^{+}(\pi)\cup\operatorname{inv}^{-}(\pi)\cup\operatorname{inv}^{\circ}(\pi),$
$\displaystyle\operatorname{inv_{D_{N}}}(\pi)$
$\displaystyle:=\operatorname{inv}^{+}(\pi)\cup\operatorname{inv}^{-}(\pi).$

We also refer to [25, Section 2] for a discussion of more general
$d$-inversions in the classical types of Coxeter groups.

The numbers of elements in $W_{N}$, $W\in\\{A,B,D\\}$, with exactly
$k\in\\{0,1,\ldots,N\\}$ inversions are called the $W_{N}$-Mahonian numbers.
If we define a random variable $X_{N}:=X(W_{N})$ as the number of inversions
of an element of $W_{N}$ chosen uniformly at random, the distribution of
$X_{N}$ is called the $W_{N}$-Mahonian distribution. The generating polynomial
for $X_{N}$ is given by

(6) $P_{N}(z)=\prod_{1\leq j\leq
N}\frac{1-z^{d_{j}}}{1-z}\qquad(z\in\mathbb{R}),$

where $d_{1},\ldots,d_{N}$ are the degrees of $W_{N}$, see [8, Theorem 7.1.5]
and note that the exponents $e_{j}$ of Coxeter groups therein are related to
the degrees by $d_{j}=e_{j}+1$. The degrees of the three types of Coxeter
groups we consider are summarized in the following table.

Type | Degrees
---|---
$A_{N}$ | $d_{1}=2,d_{2}=3,\dots,d_{N}=N+1$
$B_{N}$ | $d_{1}=2,d_{2}=4,\dots,d_{N}=2N$
$D_{N}$ | $d_{1}=2,d_{2}=4,\dots,d_{N-1}=2N-2,d_{N}=N$

In particular, the polynomials (6) are of the form (4) with $b_{j}=d_{j}$, so
that $M_{N}=\max_{1\leq j\leq N}d_{j}$, and $a_{j}=1$ for
$j\in\\{1,\ldots,N\\}$, and have degree $n=\sum_{1\leq j\leq N}(d_{j}-1)$. We
refer to Figure 3 for an illustration of the zero set in the case
$W_{N}=A_{N}$.

The cumulants of the corresponding random variables $X_{N}$ admit the
representation (5). For instance, the variance of $X_{N}$ is given by

$\sigma_{N}^{2}=\frac{1}{12}\sum_{1\leq j\leq N}(d_{j}^{2}-1).$

For each of the three possible choices for $W$ we consider, the respective
results for $\sigma_{N}^{2}$, $M_{N}$ and $\kappa_{4,N}^{\ast}$ are summarized
in the following table.

Type | $\sigma_{N}^{2}$ | $M_{N}$ | $\kappa_{4,N}^{*}$
---|---|---|---
$A_{N}$ | $\frac{1}{72}\left(2N^{3}+9N^{2}+7N\right)$ | $N+1$ | $-\frac{36}{25}\frac{6N^{5}+45N^{4}+130N^{3}+180N^{2}+89N}{4N^{6}+36N^{5}+109N^{4}+126N^{3}+49N^{2}}$
$B_{N}$ | $\frac{1}{36}\left(4N^{3}+6N^{2}-N\right)$ | $2N$ | $-\frac{18}{25}\frac{48N^{5}+120N^{4}+80N^{3}-23N}{16N^{6}+48N^{5}+28N^{4}-12N^{3}+N^{2}}$
$D_{N}$ | $\frac{1}{36}\left(4N^{3}-3N^{2}-N\right)$ | $2N-2$ | $-\frac{18}{25}\frac{48N^{4}-105N^{3}+80N^{2}-23}{16N^{5}-24N^{4}+N^{3}+6N^{2}+N}$

For all three types $A_{N}$, $B_{N}$ and $D_{N}$ this leads to expressions for
$\Delta_{N}$:

$\displaystyle\Delta_{N}=\begin{cases}\frac{\pi^{2}}{12}\sqrt{\frac{7}{3}}\frac{(2N^{3}+9N^{2}+7N)^{1/2}}{N+1}&:\text{type
}A_{N}\\\
\frac{\pi^{2}}{12}\sqrt{\frac{7}{6}}\frac{(4N^{3}+6N^{2}-N)^{1/2}}{N}&:\text{type
}B_{N}\\\
\frac{\pi^{2}}{12}\sqrt{\frac{7}{6}}\frac{(4N^{3}-3N^{2}-N)^{1/2}}{N-1}&:\text{type
}D_{N}.\end{cases}$

Hence, for these three types of Coxeter groups, we observe a central limit
theorem for $X_{N}$ with speed of convergence of order
$\mathcal{O}(N^{-1/2})$. This agrees with the results found in [25, Theorem
2.8, Corollary 2.9], where generalized inversions in finite Weyl groups are
studied. Figure 4 shows the number of inversions in a sample of $350.000$
random permutations of $N=50$ elements (remind that the Coxeter group of Type
$A_{N}$ corresponds to the symmetric group on $\\{1,\dots,N+1\\}$). Also, we
have a moderate deviation principle as in Corollary 2.3 and a concentration
inequality as given in Corollary 2.5. In addition, using $\kappa_{4,N}^{\ast}$
as given in the table above, we get
$\kappa_{4,N}^{\ast}\Delta_{N}^{2}\to-7\pi^{4}/100$ as $N\to\infty$,
independently of the type $W\in\\{A,B,D\\}$. Thus,
$\sqrt{\Delta_{N}}X_{N}^{\ast}$ converges mod-Gaussian in the sense of
Corollary 2.4 with limit function $\exp(-7\pi^{4}z^{4}/2400)$. We note in this
context that in Remark 6.9 in [22] the authors already mention without proof
and without giving details a possible mod-Gaussian convergence for the
sequence of random variables $X_{N}$.

Figure 4. Simulation of the number of inversions of a permutation of $N=50$
elements, drawn from $350.000$ samples.

####  4.3.2. Gaussian polynomials

Let $p(N,\ell,j)$ denote the number of partitions of an integer $j$ into at
most $\ell$ summands, each of which is less than or equal to $N$ (in
particular, $p(N,\ell,N\ell)=1$ and $p(N,\ell,j)=0$ if $j>N\ell$). According
to [3, Theorem 3.1], the corresponding generating polynomial $G(N,\ell;z)$ is
given by

(7) $G(N,\ell;z)=\sum_{0\leq j\leq N\ell}p(N,\ell,j)z^{j}=\prod_{1\leq j\leq
N}\frac{1-z^{j+\ell}}{1-z^{j}}\qquad(z\in\mathbb{R}).$

These polynomials were historically first studied by Gauss and hence are
called Gaussian polynomials.

The right hand side of (7) matches (4) with $b_{j}=\ell+j$ and $a_{j}=j$, and
thus $G(N,\ell;z)$ is a polynomial of degree $n=\sum_{1\leq j\leq
N}\ell=N\ell$ (note the additional dependence on the parameter $\ell$). By
(5), the cumulants of the corresponding random variables $X_{N,\ell}$ are
given by

(8) $\displaystyle\kappa_{m,N,\ell}$ $\displaystyle=\frac{B_{m}}{m}\sum_{1\leq
j\leq N}\big{(}(j+\ell)^{m}-j^{m}\big{)}$
$\displaystyle=\frac{B_{m}}{m}\left(-H_{N}^{(-m)}-\zeta(-m,\ell+N+1)+\zeta(-m,\ell+1)\right),$

where $H_{N}^{(-m)}=\sum_{0\leq k\leq N}k^{m}$ is the $N$-th harmonic number
of order $-m$ and $\zeta(-m,a)$ denotes the Hurwitz zeta function, which for
any $a\in\mathbb{R}$ is given by

$\zeta(-m,a)=-\frac{\mathcal{B}_{m+1}(a)}{m+1},$

with $\mathcal{B}_{m}(\,\cdot\,)$ denoting the $m$-th Bernoulli polynomial. In
case of the variance, this leads to

$\displaystyle\sigma_{N,\ell}^{2}$
$\displaystyle=\frac{1}{12}\left(-\frac{N^{3}}{3}-\frac{N^{2}}{2}-\frac{N}{6}-\zeta(-2,\ell+N+1)+\zeta(-2,\ell+1)\right)$
$\displaystyle=\frac{1}{12}\left(-\frac{N^{3}}{3}-\frac{N^{2}}{2}-\frac{N}{6}+\frac{\mathcal{B}_{3}(\ell+N+1)}{3}-\frac{\mathcal{B}_{3}(\ell+1)}{3}\right).$

The Bernoulli polynomial $\mathcal{B}_{3}(a)$ is given by
$\mathcal{B}_{3}(a)=a^{3}-\frac{3}{2}a^{2}+\frac{1}{2}a$, see for example [1,
Chapter 23] for details. Thus,

$\sigma_{N,\ell}^{2}=\frac{1}{12}\big{(}\ell^{2}N+\ell N+\ell N^{2}\big{)}.$

Further, $M_{N,\ell}=\max_{1\leq j\leq N}(j+\ell)=N+\ell$, so that

$\frac{M_{N,\ell}^{2}}{\sigma_{N,\ell}^{2}}=\frac{12(\ell+N)^{2}}{\ell
N(\ell+N+1)},$

which directly leads to

$\displaystyle\Delta_{N,\ell}=\pi^{2}\sqrt{\frac{7}{6}}\;\frac{\sigma_{N,\ell}}{M_{N,\ell}}=\frac{\pi^{2}}{6}\sqrt{\frac{7}{2}}\;\frac{\sqrt{\ell
N(\ell+N+1)}}{(\ell+N)},$

with $\Delta_{N,\ell}$ as in (3.1), taking into account the additional
dependence on $\ell$. As we take both $N\to\infty$ and $\ell\to\infty$, we
have $\sigma_{N,\ell}/M_{N,\ell}\to\infty$. In particular, Corollary 2.2
yields a central limit theorem for the random variables $X_{N,\ell}$ as
previously shown in [24, Section 4] and [37, Section 3]. Moreover, the speed
of convergence is bounded by

$\displaystyle\sup_{x\in\mathbb{R}}|{\mathbb{P}}(X_{N,\ell}^{\ast}\geq
x)-{\mathbb{P}}(Z\geq x)|\leq c\big{(}N^{-1/2}+\ell^{-1/2}\big{)},$

where $c>0$ is some absolute constant and $Z$ is a standard random variable.
To show mod-Gaussian convergence, using (8) as well as
$\mathcal{B}_{5}(a)=a^{5}-{5\over 2}a^{4}+{5\over 3}a^{3}-{1\over 6}a$, we
find

$\displaystyle\kappa_{4,N,\ell}=-\frac{1}{120}\left(-\frac{N^{5}}{5}-\frac{N^{4}}{2}-\frac{N^{3}}{3}+\frac{N}{30}+\frac{\mathcal{B}_{5}(\ell+N+1)}{5}-\frac{\mathcal{B}_{5}(\ell+1)}{5}\right),$

which leads to

$\kappa_{4,N,\ell}^{\ast}=-\frac{6}{5}\left(\frac{1}{N}+\frac{1}{\ell}-\frac{1}{\ell+N+1}\right),$

after straightforward simplifications. As a consequence,

$\displaystyle\kappa_{4,N,\ell}^{\ast}\Delta_{N,\ell}^{2}$
$\displaystyle=-\frac{6}{5}\left(\frac{1}{N}+\frac{1}{\ell}-\frac{1}{\ell+N+1}\right)\cdot\frac{7\pi^{4}}{72}\frac{\ell
N(\ell+N+1)}{(\ell+N)^{2}}$
$\displaystyle=-\frac{35\pi^{4}}{432}\Big{(}1+\frac{\ell+N-\ell
N}{(\ell+N)^{2}}\Big{)}\longrightarrow-\frac{35\pi^{4}}{432}K,$

as both $N\to\infty$ and $\ell\to\infty$, where $K\in\mathbb{R}$ is the limit
of $1+(\ell+N-\ell N)/(N+\ell)^{2}$ as $N,\ell\to\infty$. For instance, if
$\ell=N$, we have $K=3/4$. As a result, Corollary 2.4 (ii) with
$L=-35\pi^{4}K/432$ gives mod-Gaussian convergence of
$\sqrt{\Delta_{N,\ell}}X_{N}^{\ast}$. It appears that the results which follow
from Section 1 (especially the Berry-Esseen bound and the mod-Gaussian
convergence) seem new. In particular, note that for Gaussian polynomials the
generating function does not factorize as required in [18, Lemma 4], so that
the latter result cannot be applied in this situation in order to deduce a
Berry-Esseen bound for the coefficients of a Gaussian polynomial.

####  4.3.3. Catalan numbers

There are several ways of generalizing the usual Catalan numbers
$C_{N}=\frac{1}{N+1}\binom{2N}{N}$ to $q$-Catalan numbers. One of them reads

$C_{N}(q)=\frac{1}{[N+1]_{q}}\left[{2N}\atop{N}\right]_{q},$

where $[N]_{q}:=1+q+\dots+q^{N-1}$ and

$\left[{{N}\atop{k}}\right]_{q}:=\frac{[N]_{q}!}{[k]_{q}![N-k]_{q}!},$

with $[N]_{q}!:=[N]_{q}[N-1]_{q}\cdots[1]_{q}$. For a broader view on
$q$-Catalan numbers we refer the reader to [14]. In the sequel, we assume
$N\geq 2$. It is easy to check that

$C_{N}(q)=\prod_{2\leq j\leq N}\frac{1-q^{N+j}}{1-q^{j}}.$

which shows that $q$-Catalan numbers are closely related to the Gaussian
polynomials (7). Indeed, $C_{N}(q)=G(N,N;q)/[N+1]_{q}$. In particular,
$C_{N}(q)$ is a polynomial as in (4) with $b_{1}=1$, $a_{1}=1$, $b_{j}=N+j$
and $a_{j}=j$ for $2\leq j\leq N$. Hence, the $m$-th cumulant of the
corresponding random variable $X_{N}$ is given by

(9) $\kappa_{m,N}=\frac{B_{m}}{m}\sum_{2\leq j\leq
N}\big{(}(N+j)^{m}-j^{m}\big{)}.$

It follows that

$\sigma_{N}^{2}=\frac{1}{6}\big{(}N^{3}-N\big{)},$

and since $M_{N}=\max_{1\leq j\leq N}b_{j}=2N$, we have

$\frac{M_{N}^{2}}{\sigma_{N}^{2}}=\frac{24N}{N^{2}-1}.$

Therefore,

$\Delta_{N}=\pi^{2}\sqrt{\frac{7}{6}}\frac{\sigma_{N}}{M_{N}}=\frac{\pi^{2}\sqrt{7}}{12}\sqrt{\frac{N^{2}-1}{N}}.$

Hence, Corollary 2.2 yields a central limit theorem for the sequence
$(X_{N}^{\ast})_{N\geq 1}$ as previously also shown in [10, Corollary 3.3],
see also [7, Theorem 3.1]. Using Corollary 2.2, the speed of convergence in
this central limit theorem is bounded by

$\sup_{x\in\mathbb{R}}|{\mathbb{P}}(X_{N}^{\ast}\geq x)-{\mathbb{P}}(Z\geq
x)|\leq cN^{-1/2},$

where $c>0$ is some absolute constant and $Z$ a standard normal random
variable. This answers the question for a Berry-Esseen bound for $q$-Catalan
numbers which was raised in [38, Remark 4.2]. Note that moreover, Corollary
2.3 recovers the moderate deviation principle established in [38, Theorem
2.1], while the remaining results which follow from Theorem 2.2 seem new. In
particular, noting that by (9),

$\kappa_{4,N}=-\frac{1}{60}\big{(}3N^{5}+3N^{4}-N^{3}-3N^{2}-2N\big{)},\qquad\kappa_{4,N}^{\ast}=-\frac{3}{5}\Big{(}\frac{3N^{2}+3N+2}{N^{3}-N}\Big{)},$

we obtain

$\kappa_{4,N}^{\ast}\Delta_{N}^{2}=-\frac{7\pi^{4}}{270}\frac{3N^{2}+3N+2}{N^{2}}\longrightarrow-\frac{7\pi^{4}}{80}$

as $N\to\infty$. By Corollary 2.4 (ii), we thus obtain that
$\sqrt{\Delta_{N}}X_{N}^{\ast}$ converges in the mod-Gaussian sense with
parameters $\Delta_{N}$ and limiting function $\exp(-7\pi^{4}z^{4}/1920)$.

######  Remark 4.2.

Generalizing the usual Catalan numbers $C_{N}$ in another way, one may define
for integers $k\geq 1$ the $k$-Catalan numbers by

$C_{N,k}=\frac{1}{(k-1)N+1}\binom{kN}{N},$

see [35]. Given the $q$-Catalan numbers, it is natural to define

$C_{N,k}(q):=\frac{1}{[(k-1)N+1]_{q}}\left[{kN}\atop{N}\right]_{q},$

using the same notation as before. It is easy to show that

$C_{N,k}(q)=\prod_{2\leq j\leq N}\frac{1-q^{(k-1)N+j}}{1-q^{j}},$

which is a polynomial of the form discussed here. Hence, generalizing the
results from from above, one can analogously show that

$\Delta_{N,k}=\sqrt{\frac{7}{6}}\;\pi^{2}\frac{\sigma_{N,k}}{M_{N,k}}=\pi^{2}\sqrt{\frac{7}{2}}\frac{\sqrt{(k-1)(N-1)N(kN+2)}}{6kN}.$

In particular, for any choice of $k>1$ by Corollary 2.2, the sequence
$(X_{N}^{\ast})_{N\geq 1}$ satisfies a central limit theorem as $N\to\infty$
as previously shown in [10, Section 3]. Noting that $\Delta_{N,k}$ can always
be chosen of order $\mathcal{O}(N^{-1/2})$, all the other results from Section
2 follow readily.

####  4.3.4. Descending plane partitions

A descending plane partition (DPP for short and first introduced by [2]) is an
array of non-negative integers $\gamma_{i,j}$ ($i\leq j$), which are called
the parts of the DPP, as in Figure 5, such that the following conditions hold:

  1. (D1)

The values of the parts are decreasing in each row from left to right and
strictly decreasing in each column from top to bottom. In particular,
$\gamma_{i,i}$ is the largest part of the $i$-th row and the $i$-th column.

  2. (D2)

The entry $\gamma_{i,i}$ is strictly greater than the number of parts in the
$i$-th row and less or equal to the number of parts in the $(i-1)$-th row.

We refer, for example, to [36, Section 1] or [29, Section 1] for a detailed
introduction. A descending plane partition is said to be of order $N$, if its
largest part is at most $N$. For instance, there are two DPPs of order $N=2$
and seven DPPs of order $N=3$. Note that $\emptyset$ always counts as a DPP
(the empty one). If $\gamma_{i,j}\leq j-i$, $\gamma_{i,j}$ is called a special
part of the DPP. See Figure 5 for a DPP of order $N=12$ (and thus also of any
order $N>12$), where the specials parts are marked in bold.

Writing $S:=\sum_{i,j}\gamma_{i,j}$ for the sum of the parts, a DPP of order
$N$ can be regarded as a partition of $S$ whose largest part is at most $N$,
see especially [36, Lemma 5] which translates (D1) and (D2) into conditions on
the partition for DPPs with no special parts. Descending plane partitions
moreover admit connections to other combinatorial structures. For instance, in
[4] a bijection between DPPs of order $N$ with no special parts and certain
permutations of $N$ elements is established. There is also a relation of DPPs
to alternating sign matrices as studied in [29, 30].

$\gamma_{1,1}$ |  $\gamma_{1,2}$ |  $\gamma_{1,3}$ |  … |  … |  … |  … |  $\gamma_{1,\lambda_{1}}$
---|---|---|---|---|---|---|---
|  $\gamma_{2,2}$ |  $\gamma_{2,3}$ |  … |  … |  … |  $\gamma_{2,\lambda_{2}}$ |
|  |  $\ddots$ |  |  |  |  $\ddots$ |
|  |  |  $\gamma_{\ell,\ell}$ |  $\dots$ |  $\gamma_{\ell,\lambda_{\ell}}$ |  |
(a) A general descending plane partition

$12$ |  $12$ |  $11$ |  $9$ |  $8$ |  $\boldsymbol{5}$ |  $\boldsymbol{1}$ |
---|---|---|---|---|---|---|---
|  $\;7$ |  $\;7$ |  $6$ |  $5$ |  $\boldsymbol{4}$ |  |
|  |  $\;5$ |  $5$ |  $3$ |  $\boldsymbol{3}$ |  |
|  |  |  $3$ |  $2$ |  |  |
(b) A descending plane partition of order $N=12$ where special parts are
marked in bold

Figure 5. Descending plane partitions

According to [36, Corollary 6], the generating polynomial for DPPs of order
$N$ with no special parts is given by

(10) $\prod_{1\leq j\leq N}[j]_{q^{j}}=\prod_{1\leq j\leq
N}\frac{1-q^{j^{2}}}{1-q^{j}},$

where we recall the notation $[j]_{q}=1+q+q^{2}+\dots+q^{j-1}$ from Section
4.3.3. This generating polynomial also appears in the context of the inversion
index of a permutation of $N$ elements as shown in [36] using permutations
matrices see also [St000616] in [32]).

The polynomial (10) matches (4) with $b_{j}=j^{2}$ and $a_{j}=j$, hence the
cumulants are given by

(11) $\kappa_{m,N}=\frac{B_{m}}{m}\sum_{1\leq j\leq
N}\big{(}j^{2m}-j^{m}\big{)},$

which directly leads to

$\sigma_{N}^{2}=\frac{1}{120}\left(2N^{5}+5N^{3}-5N^{2}-2N\right).$

Having $M_{N}=N^{2}$, it follows that

$\Delta_{N}=\frac{\pi^{2}}{12}\sqrt{\frac{7}{5}}\;\frac{\left(2N^{5}+5N^{3}-5N^{2}-2N\right)^{1/2}}{N^{2}},$

with $\Delta_{N}$ as in Theorem 3.1. Therefore, Corollary 2.2 provides a
central limit theorem for the number of DPPs with largest part at most $N$
with speed of convergence of order $\mathcal{O}(N^{-1/2})$. See Figure 6 for a
plot of $350.000$ samples with $N=50$. The Gaussian shape is clearly visible.
Further, Corollary 2.3 provides a moderate deviation principle and Corollary
2.5 a concentration inequality. Moreover, mod-Gaussian convergence follows
from Corollary 2.4 (ii). Indeed, using (11),

$\displaystyle\kappa_{4,N}$
$\displaystyle=-\frac{1}{2160}\left(2N^{9}+9N^{8}+12N^{7}-12N^{5}-9N^{4}-2N^{3}\right),$
$\displaystyle\kappa_{4,N}^{\ast}$
$\displaystyle=-\frac{20(N^{2}+N)}{6N^{3}+9N^{2}-9N-6},$

and hence,

$\kappa_{4,N}^{\ast}\Delta_{N}^{2}=-\frac{7\pi^{4}}{36}\frac{(N^{2}+N)(2N^{5}+5N^{3}-5N^{2}-2N)}{N^{4}(6N^{3}+9N^{2}-9N-6)}\longrightarrow-\frac{7\pi^{4}}{108},$

as $N\to\infty$, which implies mod-Gaussian convergence with parameter
$\Delta_{N}$ and limit function $\exp(-7\pi^{4}z^{4}/2592)$. All these results
seem new.

Figure 6. Simulation of the number of entries in a descending plane partitions
of order $N=50$, drawn from $350.000$ samples.

##  5\. Proofs

###  5.1. Proof of Theorem 2.1

The strategy is to apply the cumulant bound provided by [27, Lemma 8.1] and
its proof. To this end, we set $u_{N}(z):=u_{X_{N}}(z)=\log|{f_{N}(z)}|$ for
each $X_{N}$ as well as

$S_{R}(\delta_{N}):=\big{\\{}z\in\mathbb{C}:|{z}|\in[R^{-1},R],\arg(z)\in[-\delta_{N},\delta_{N}]\big{\\}}$

for $R>1$. By assumption, $f_{N}$ is an analytic function on the complex plane
which is zero-free in $S(\delta_{N})$, and hence, $u_{N}$ is a harmonic
function in a neighbourhood of $S_{R}(\delta_{N}/2)$ for any $R>1$.
Furthermore, $u_{N}$ is invariant under complex conjugation since

$u_{N}(\bar{z})=\log|{f_{N}(\bar{z})}|=\log|{\overline{f_{N}(z)}}|=\log|{f_{N}(z)}|=u_{N}(z),$

using that $f_{N}$ is a polynomial with real coefficients.

Moreover, we have $u_{N}(|z|)\geq u_{N}(z)$ for any $z\in S(\delta_{N})$,
since

(12)
$\log|{f_{X_{N}}(z)}|=\log|{\mathbb{E}z^{X_{N}}}|\leq\log\mathbb{E}|{z}|^{X_{N}}=\log|{f_{X_{N}}(|{z}|)}|$

(this property is called _weak positivity_ in [27]). In other words, on any
circle centred at the origin, $u_{N}$ takes its maximum on the positive real
axis. To apply [27, Lemma 8.1], one needs a sharpening of this property and
show that in a suitable neighbourhood of $1\in\mathbb{C}$, $u_{N}$ decreases
counter-clockwise (and due to the invariance under complex conjugation also
clockwise) starting at the positive real axis on any circle centred at the
origin. That is,

(13) $u(\rho e^{i\theta_{1}})\geq u(\rho e^{i\theta_{2}})$

for any $\rho>0$ and $0\leq\theta_{1}\leq\theta_{2}\leq\pi$ in some ball
around $1\in\mathbb{C}$ and such that $\rho e^{i\theta_{1}}$, $\rho
e^{i\theta_{2}}\in S(\delta_{N})$ (this property is called _$0$ -decreasing_
in [27]).

We will show (13) for $\rho e^{i\theta_{1}},\rho e^{i\theta_{2}}\in
S_{2}(\delta_{N}/2)$. By [27, Lemma 4.1], since $u_{N}$ satisfies (12), is
invariant under complex conjugation and is analytic in a neighbourhood of
$S_{R}(\delta_{N})$ for any $R>1$, a sufficient condition for (13) to hold in
$S_{2}(\delta_{N}/2)$ is given by

(14) $\Big{(}\frac{2}{R}\Big{)}^{1/\delta_{N}}\max_{\begin{subarray}{c}z\in
S_{R}(0,\delta_{N})\\\
|{z}|\in\\{R^{-1},R\\}\end{subarray}}|{u_{N}(z)}|\longrightarrow 0$

as $R\to\infty$. Noting that $f_{N}$ is a polynomial, we have
$u_{N}(z)=\mathcal{O}(\log|z|)$ and $u_{N}(1/z)=\mathcal{O}(\log|z|)$ as
$z\to\infty$. In particular, this implies that (14) holds true since its left
hand side is of order $\mathcal{O}(R^{-1/\delta_{N}}\log(R))$ as $R\to\infty$.

Therefore, we may apply [27, Lemma 8.1] with the choice
$\varepsilon=\delta_{N}/4$, noting that the condition $\sum_{j\geq
2}|{\kappa_{j}}|(j!)^{-1}(\varepsilon/32)^{j}>0$ therein is trivially
satisfied. This yields

$|{\kappa_{m}^{\ast}}|\leq{m!\over(c\delta_{N}\sigma_{N})^{m-2}}$

for all $m\geq 3$ with $c={1\over 4}\cdot 2^{-3246}=2^{-3248}$. ∎

###  5.2. Proof of the cumulant representation (5)

For the reader’s convenience and since the formula in [20] needs a correction
we derive here the cumulant representation (5) based on the methods of the
proof of [10, Theorem 2.3]. First note that the moment generating function of
the random variable $X_{N}$ is given by

$m_{X_{N}}(t):=\mathbb{E}(e^{tX_{N}})=\frac{P_{N}(e^{t})}{P_{N}(1)}=\prod_{1\leq
j\leq N}\frac{1-e^{tb_{j}}}{1-e^{ta_{j}}}\cdot\frac{a_{j}}{b_{j}},$

recalling that $P_{N}(1)=\prod_{1\leq j\leq N}b_{j}/a_{j}$ (see the discussion
following (4)). Using $2\sinh(x)=e^{x}-e^{-x}$ then leads to

$\displaystyle m_{X_{N}}(t)$ $\displaystyle=\prod_{1\leq j\leq
N}e^{t(b_{j}-a_{j})/2}\frac{\sinh(tb_{j}/2)/(b_{j}/2)}{\sinh(ta_{j}/2)/(a_{j}/2)}$
$\displaystyle=\exp\Big{(}\frac{t}{2}\sum_{1\leq j\leq
N}(b_{j}-a_{j})\Big{)}\cdot\prod_{1\leq j\leq
N}\frac{\sinh(tb_{j}/2)/(b_{j}/2)}{\sinh(ta_{j}/2)/(a_{j}/2)}$

and hence,

$\displaystyle\ln\big{(}m_{X_{N}}(t)\big{)}$
$\displaystyle=\frac{t}{2}\sum_{1\leq j\leq N}(b_{j}-a_{j})$
$\displaystyle\qquad+\sum_{1\leq j\leq
N}\Big{(}\ln\Big{(}\frac{\sinh(tb_{j}/2)}{b_{j}/2}\Big{)}-\ln\Big{(}\frac{\sinh(ta_{j}/2)}{a_{j}/2}\Big{)}\Big{)}.$

By [33, Chapter 1.3.1],

$\ln\Big{(}\frac{\sinh(x/2)}{x/2}\Big{)}=\sum_{m\geq
1}B_{2m}\frac{x^{2m}}{2m(2m)!},$

with $B_{m}$ being the Bernoulli numbers. This yields

$\ln\big{(}m_{X_{N}}(t)\big{)}=\frac{t}{2}\sum_{1\leq j\leq
N}(b_{j}-a_{j})+\sum_{m\geq 1}B_{2m}\frac{t^{2m}}{2m(2m)!}\sum_{1\leq j\leq
N}(b_{j}^{2m}-a_{j}^{2m}).$

By definition, the cumulants of $X_{N}$ are given as the coefficients of

$m_{X_{N}}(t)=\exp\bigg{(}\sum_{m\geq 1}\frac{\kappa_{m}}{m!}t^{m}\bigg{)},$

which by comparison leads to

$\kappa_{2m,N}=\frac{B_{2m}}{2m}\sum_{1\leq j\leq N}(b_{j}^{2m}-a_{j}^{2m})$

and $\kappa_{2m+1,N}=0$. Moreover,

$\kappa_{1,N}=\frac{1}{2}\sum_{1\leq j\leq N}(b_{j}-a_{j})=B_{1}\sum_{1\leq
j\leq N}(b_{j}-a_{j})$

for the choice of $B_{1}=\frac{1}{2}$ (the sign of $B_{1}$ is a matter of
convention). In particular, we see that $\kappa_{1,N}=\mathbb{E}[X_{N}]=n/2$
in accordance with [20, Lemma 2.2 & Equation (20)]. ∎

###  5.3. Proof of Theorem 3.1

######  Lemma 5.1.

For any two real numbers $b\geq a\geq 0$ and any integer $m\geq 2$, it holds
that

(15) $b^{2m}-a^{2m}\leq(b^{2}-a^{2})2^{m-1}b^{2m-2}.$

###### Proof.

First note, that if $a=b$, (15) is trivial. In the case $b>a$, the statement
can be proven by induction over $m$. The case $m=2$ is clear and we are left
to show that if (15) holds for some $m\in\mathbb{N}$, then also

$b^{2m+2}-a^{2m+2}\leq(b^{2}-a^{2})2^{m}b^{2m}.$

Since $b\geq a$, we obtain

$\frac{b^{2m+2}-a^{2m+2}}{b^{2m}-a^{2m}}=b^{2}+\frac{a^{2m}b^{2}-a^{2m+2}}{b^{2m}-a^{2m}}\leq
b^{2}+\frac{a^{2}b^{2m}-a^{2m+2}}{b^{2m}-a^{2m}}\leq 2b^{2},$

which yields

$\displaystyle b^{2m+2}-a^{2m+2}$
$\displaystyle\leq(b^{2m}-a^{2m})2b^{2}\leq(b^{2}-a^{2})2^{m}b^{2m},$

where the last step follows by induction. ∎

###### Proof of Theorem 3.1.

For $m\geq 2$, the cumulant representation (5), the identity
$\kappa_{2m,N}^{\ast}=\kappa_{2m,N}/\sigma_{N}^{2m}$ and our assumption that
$b_{j}\geq a_{j}$ for all $j\in\\{1,\ldots,N\\}$ give

$|\kappa_{2m,N}^{\ast}|=\frac{|B_{2m}|}{2m\sigma_{N}^{2m}}\sum_{1\leq j\leq
N}(b_{j}^{2m}-a_{j}^{2m}).$

An application of Lemma 5.1 then leads to

$\displaystyle|\kappa_{2m,N}^{\ast}|$
$\displaystyle\leq\frac{|B_{2m}|}{2m\sigma_{N}^{2m}}\sum_{1\leq j\leq
N}2^{m-1}b_{j}^{2m-2}(b_{j}^{2}-a_{j}^{2})$
$\displaystyle\leq\frac{2^{m-2}}{m\sigma_{N}^{2m}}|B_{2m}|M_{N}^{2m-2}\sum_{1\leq
j\leq N}(b_{j}^{2}-a_{j}^{2}),$

where we recall that $M_{N}=\max_{1\leq j\leq N}b_{j}$ as stated in the
theorem. Again by (5), the variance can be written as

$\sigma_{N}^{2}=\kappa_{2,N}=\frac{B_{2}}{2}\sum_{1\leq j\leq
N}(b_{j}^{2}-a_{j}^{2}).$

Noting that $B_{2}=1/6$, this yields

$|\kappa_{2m,N}^{\ast}|\leq\frac{12\cdot
2^{m-2}}{m}|B_{2m}|\Big{(}\frac{M_{N}}{\sigma_{N}}\Big{)}^{2m-2}.$

To bound the Bernoulli numbers $B_{2m}$ we use [1, Equation 3.1.15], which
states that

$|B_{2m}|\leq\frac{2(2m)!}{(2\pi)^{2m}}\frac{1}{1-2^{1-2m}}\qquad(m\geq 1).$

As a consequence,

$\displaystyle|\kappa_{2m,N}^{\ast}|$ $\displaystyle\leq\frac{12\cdot
2^{m-2}}{m}\frac{2(2m)!}{(2\pi)^{2m}}\frac{1}{1-2^{1-2m}}\Big{(}\frac{M_{N}}{\sigma_{N}}\Big{)}^{2m-2}$
$\displaystyle=(2m)!\frac{3\cdot
2^{1-m}}{m\pi^{2m}(1-2^{1-2m})}\Big{(}\frac{M_{N}}{\sigma_{N}}\Big{)}^{2m-2}.$

The sequence $(c_{m})_{m\geq 2}$ defined by $c_{m}:=\frac{3\cdot
2^{1-m}}{m\pi^{2m}(1-2^{1-2m})}$ is strictly decreasing, since

$c_{m}-c_{m+1}=3\pi^{-2m}2^{m+1}\left(\frac{1}{\left(4^{m}-2\right)m}-\frac{1}{\pi^{2}\left(2^{2m+1}-1\right)(m+1)}\right)>0.$

To see this, note that for $m\geq 2$,

$\displaystyle\pi^{2}(2^{2m+1}-1)(m+1)$
$\displaystyle=m(4^{m}+1)\pi^{2}+(2^{2m+1}-1)\pi^{2}>m(4^{m}-2),$

and therefore,

$\displaystyle\frac{1}{m(4^{m}-2)}>\frac{1}{(m+1)(2^{2m+1}-1)\pi^{2}}.$

In particular, this implies that $\max_{m\geq 2}c_{m}=c_{2}=6/(7\pi^{4})$.
Thus,

$\displaystyle|\kappa_{2m,N}^{\ast}|$
$\displaystyle\leq(2m)!\frac{6}{7\pi^{4}}\Big{(}\frac{M_{N}}{\sigma_{N}}\Big{)}^{2m-2}.$

So, choosing

$\Delta_{N}=\max_{m\geq
2}\Big{\\{}\sqrt[2m-2]{\frac{7\pi^{4}}{6}}\Big{\\}}\frac{\sigma_{N}}{M_{N}}=\sqrt{\frac{7\pi^{4}}{6}}\frac{\sigma_{N}}{M_{N}}$

we arrive at

$|\kappa_{2m,N}^{\ast}|\leq\frac{(2m)!}{\Delta_{N}^{2m-2}}.$

This completes the proof. ∎

##  Appendix A The method of cumulants

In order to keep this paper self-contained, we briefly summarize the
probabilistic consequences which can be drawn from the Statulevičius
condition. This approach is known as the method of cumulants and we refer to
the monograph [34] as well as the recent survey [12] for a detailed account of
this method. Recall that for a sequence of random variables $(X_{N})_{N\geq
1}$, we denote the cumulant of order $m$ of $X_{N}$ by
$\kappa_{m,N}:=\kappa_{m}(X_{N})$ and the respective cumulant of the
normalization
$X_{N}^{\ast}:=(X_{N}-\mathbb{E}X_{N})/\sqrt{\operatorname{var}(X_{N})}$ by
$\kappa_{m,N}^{\ast}:=\kappa_{m}(X_{N}^{\ast})$. We present the result in a
more general form, although we had $\gamma=0$ in all our applications.

######  Lemma A.1.

Let $(X_{N})_{N\geq 1}$ be a sequence of random variables which satisfies

(16) $|{\kappa_{m,N}^{\ast}}|\leq\frac{(m!)^{1+\gamma}}{\Delta_{N}^{m-2}}$

for all $m\geq 3$, all $N\geq 1$, some $\gamma\geq 0$ and $\Delta_{N}>0$.
Assume that $\Delta_{N}\to\infty$ as $N\to\infty$.

  * (i)

For any $N\geq 1$ it holds that

$\sup_{x\in\mathbb{R}}|{{\mathbb{P}}(X_{N}\geq x)-{\mathbb{P}}(Z\geq
x)}|\leq\frac{C_{\gamma}}{\Delta_{N}^{1/(1+2\gamma)}},$

where $C_{\gamma}\in(0,\infty)$ only depends on $\gamma$.

  * (ii)

For every sequence $(a_{N})_{N\geq 1}$ with $a_{N}\to\infty$ and
$a_{N}=o(\Delta_{N}^{1/(1+2\gamma)})$, the sequence of random variables
$X_{N}/a_{N}$ satisfies a moderate deviation principle with speed $a_{N}^{2}$
and rate function $I(x)=x^{2}/2$.

  * (iii)

Assume that
$|\kappa_{m,N}^{\ast}|\leq(m!/2)^{1+\gamma}H/\bar{\Delta}_{N}^{m-2}$ for all
$m\geq 3$, $N\geq 1$, some $\gamma\geq 0$ and $H,\bar{\Delta}_{N}>0$. Then,
there exists a constant $C>0$ such that for every $N\geq 1$ and all $x\geq 0$,

${\mathbb{P}}(X_{N}\geq x)\leq
C\exp\Big{(}-\frac{1}{2}\frac{x^{2}}{H+x^{2-\alpha}/\bar{\Delta}_{N}^{\alpha}}\Big{)},$

where $\alpha:=1/(1+\gamma)$.

Statement (i) of the above lemma corresponds to Theorem 2.4 in [12] and to
Corollary 2.1 in [34]. The latter gives the precise value

$C_{\gamma}=2^{\frac{1}{2(2\gamma+1)}+2}\cdot 3^{\frac{1}{2\gamma+1}+3}$

for the constant. Statement (ii) agrees with Theorem 3.1 in [12] and with
Theorem 1.1 in [11]. Finally, (iii) corresponds to Theorem 2.5 in [12] and to
Corollary 2.3 in [34]. In our applications, we have set $H=2$, in which case
(together with $\gamma=0$) the assumption on the cumulants agrees with (16).

If $\gamma=0$, (16) also gives rise to mod-Gaussian convergence. We state a
corresponding result in two versions, one of them directly taken from the
literature and the other one adapted to the situation we consider in the
present paper.

######  Lemma A.2.

Let $(X_{N})_{N\geq 1}$ be a sequence of random variables.

  * (i)

Assume that $|\kappa_{m,N}|\leq(Cm)^{m}\alpha_{N}\beta_{N}^{m}$ for all $m\geq
2$ and for some sequences $(\alpha_{N})_{N\geq 1}\to\infty$ and
$(\beta_{N})_{N\geq 1}$ satisfying $\alpha_{N}\to\infty$. Further assume that
there exists an integer $v\geq 3$ such that $\kappa_{m,N}=0$ for any
$m=3,\ldots,v-1$ and that

$\displaystyle\kappa_{2,N}$
$\displaystyle=\sigma^{2}\alpha_{N}\beta_{N}^{2}(1+o(\alpha_{N}^{-(v-2)/v})),$
$\displaystyle\kappa_{v,N}$ $\displaystyle=L\alpha_{N}\beta_{N}^{v}(1+o(1)).$

Then, the sequence of random variables
$(X_{N}-\mathbb{E}[X_{N}])/(\alpha_{N}^{1/v}\beta_{N})$ converges in the mod-
Gaussian sense with parameters $t_{N}=\sigma^{2}\alpha_{N}^{(v-2)/v}$ and
limiting function $\psi(z)=e^{\frac{L}{v!}z^{v}}$.

  * (ii)

Assume that (16) holds. Further assume that there exists an integer $v\geq 3$
such that $\kappa_{m,N}^{\ast}=0$ for any $m=3,\ldots,v-1$ and that
$\kappa_{v,N}^{\ast}\Delta_{N}^{v-2}\to L\in\mathbb{R}$ as $N\to\infty$. Then,
the sequence of random variables $\Delta_{N}^{1-2/v}X_{N}^{\ast}$ converges in
the mod-Gaussian sense with parameters $\Delta_{N}^{2(v-2)/v}$ and limiting
function $e^{\frac{L}{v!}z^{v}}$.

Part (i) of Lemma A.1 is a summary of the results of Section 5.1 in [13], and
(ii) immediately follows from it as we may choose $C=1$,
$\alpha_{N}=\Delta_{N}^{2}$, $\beta_{N}=\sigma_{N}/\Delta_{N}$ and
$\sigma^{2}=1$.

### Acknowledgment

We are grateful to Martina Juhnke-Kubitzke, Kathrin Meier, Benedikt Rednoß and
Christian Stump for stimulating discussions about the content of this paper.
We would moreover like to thank an anonymous reviewer of the first version of
this paper who pointed us to the cumulant bound (2) given in [27], largely
generalizing our initial results for the class of polynomials discussed in
Section 3.
The last author was supported by the German Research Foundation (DFG) via the
priority program SPP 2265 and the collaborative research centre CRC/TRR 191.

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