Central limits from generating functions

TL;DR

This paper establishes a general framework for central limit theorems using generating functions with meromorphic properties, recovering classical results and proving a 2020 conjecture about permutation statistics.

Contribution

It introduces a new approach linking generating functions' meromorphic extensions to weak convergence to normal distributions, unifying classical CLT and permutation results.

Findings

Generalized CLT from generating functions with meromorphic extensions

Recovery of classical Lindeberg–Lévy CLT for i.i.d. sums

Proof of the 2020 permutation des statistic conjecture

Abstract

Let $(Y_n)_n$ be a sequence of $\mathbb{R}^d$-valued random variables. Suppose that the generating function \[f(x, z) = \sum_{n = 0}^\infty \varphi_{Y_n}(x) z^n,\] where $\varphi_{Y_n}$ is the characteristic function of $Y_n$, extends to a function on a neighborhood of $\{0\} \times \{z : |z| \leq 1\} \subset \mathbb{R}^d \times \mathbb{C}$ which is meromorphic in $z$ and has no zeroes. We prove that if $1 / f(x, z)$ is twice differentiable, then there exists a constant $\mu$ such that the distribution of $(Y_n - \mu n) / \sqrt{n}$ converges weakly to a normal distribution as $n \to \infty$. If $Y_n = X_1 + \cdots + X_n$, where $(X_n)_n$ are i.i.d. random variables, then we recover the classical (Lindeberg$\unicode{x2013}$L\'evy) central limit theorem. We also prove the 2020 conjecture of Defant that if $\pi_n \in \mathfrak{S}_n$ is a uniformly random permutation, then the distribution of $(\operatorname{des} (s(\pi_n)) + 1 - (3 - e) n) / \sqrt{n}$ converges, as $n \to \infty$, to a normal distribution with variance $2 + 2e - e^2$.