Central limits from generating functions

Mitchell Lee

Published 2024-06-25Version 1

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 $2 + 2e - e^2$.