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Quantification of the Fourth Moment Theorem for Cyclotomic Generating Functions

Benedikt Rednoß, Christoph Thäle

TL;DR

The paper addresses quantitative normal approximation for random variables $X_n$ supported on $\{0,\\dots,n\\}$ via their probability generating functions, under two root-location regimes: real-rooted and cyclotomic. It uses an elementary Stein–Tikhomirov approach to derive Berry-Esseen type bounds for the standardized variable $(X_n-\\mu_n)/\\sigma_n$, with a bound of order $1/\\sigma_n$ in the real-rooted case and a bound of order $|\\kappa_n|^{1/4}/\\sigma_n$ in the cyclotomic case, where $\\kappa_n$ is the fourth cumulant. The method yields explicit constants $c_1$ and $c_2$ and connects cumulants to the geometry of the roots via a detailed analysis of the function $q(t)$ derived from the roots. This provides an elementary, unified route to quantitative CLTs for a broad class of combinatorial-generating functions, matching known necessary and sufficient conditions in the real-rooted and cyclotomic settings.

Abstract

This paper deals with sequences of random variables $X_n$ only taking values in $\{0,\ldots,n\}$. The probability generating functions of such random variables are polynomials of degree $n$. Under the assumption that the roots of these polynomials are either all real or all lie on the unit circle in the complex plane, a quantitative normal approximation bound for $X_n$ is established in a unified way. In the real rooted case the result is classical and only involves the variances of $X_n$, while in the cyclotomic case the fourth cumulants or moments of $X_n$ appear in addition. The proofs are elementary and based on the Stein-Tikhomirov method.

Quantification of the Fourth Moment Theorem for Cyclotomic Generating Functions

TL;DR

The paper addresses quantitative normal approximation for random variables supported on via their probability generating functions, under two root-location regimes: real-rooted and cyclotomic. It uses an elementary Stein–Tikhomirov approach to derive Berry-Esseen type bounds for the standardized variable , with a bound of order in the real-rooted case and a bound of order in the cyclotomic case, where is the fourth cumulant. The method yields explicit constants and and connects cumulants to the geometry of the roots via a detailed analysis of the function derived from the roots. This provides an elementary, unified route to quantitative CLTs for a broad class of combinatorial-generating functions, matching known necessary and sufficient conditions in the real-rooted and cyclotomic settings.

Abstract

This paper deals with sequences of random variables only taking values in . The probability generating functions of such random variables are polynomials of degree . Under the assumption that the roots of these polynomials are either all real or all lie on the unit circle in the complex plane, a quantitative normal approximation bound for is established in a unified way. In the real rooted case the result is classical and only involves the variances of , while in the cyclotomic case the fourth cumulants or moments of appear in addition. The proofs are elementary and based on the Stein-Tikhomirov method.
Paper Structure (7 sections, 1 theorem, 45 equations, 2 figures)

This paper contains 7 sections, 1 theorem, 45 equations, 2 figures.

Table of Contents

  1. Introduction and Results
  2. Preliminary Computations
  3. Proof of Theorem \ref{['thm']}
  4. Common Ground
  5. Specialization
  6. Proof of Theorem \ref{['thm']} \ref{['thm:i']}
  7. Proof of Theorem \ref{['thm']} \ref{['thm:ii']}

Key Result

Theorem 1

For $n\in\mathbb{N}$ let $X_n$ be a random variable only taking values in $\{ 0, \dots , n \}$. Let $\mu_n=\mathbb{E} X_n$, $\sigma_n^2=\operatorname{Var}(X_n)>0$, $\kappa_n=\operatorname{Cum}_4(X_n)$ and $g_n$ be the generating function of $X_n$. Then, there are absolute constants $c_1>0$ and $c_2>

Figures (2)

  • Figure 1: Roots of the generating function $g$. The angle parameter $\delta$ is chosen such that no root (green dots) of $g$ is in the interior of the cone pointed at the origin and having central angle $\delta$ (red). Evaluation of $g$ around the unit circle (blue) yields the characteristic function $\Psi$. Note that we currently do not necessarily assume $g$ to be real rooted or cyclotomic as in \ref{['case:realrooted']} or \ref{['case:cyclotomic']}, respectively.
  • Figure 2: Behaviour of $\frac{-\varrho}{(1-\varrho)^2}$; $x$ denotes $\operatorname{Re}(\varrho)$, $y$ denotes $\operatorname{Im}(\varrho)$. According to \ref{['sigma-re']}, the real part of $\frac{-\varrho}{(1-\varrho)^2}$ is positive or zero for all $\varrho$ in the green area. According to \ref{['sigma-im']}, the imaginary part of $\frac{-\varrho}{(1-\varrho)^2}$ is zero for all $\varrho$ on the blue line and the blue circle. The orange curve is the right strophoid of the curve given by $x=2$ with respect to the pole $0+0i$ and the fixed point $1+0i$, see page 96 in Lo61.

Theorems & Definitions (2)

  • Theorem 1
  • Remark 2