Central limits from generating functions

Abstract

Let $(Y_n{)}_n$ be a sequence of ${\mathbb{R}}^d$-valued random variables. Suppose that the generating function \[f(x, z) = \sum_{n = 0}^\infty φ_{Y_n}(x) z^n,\] where $\ddot{I}{†}_{Y_n}$ is the characteristic function of $Y_n$, extends to a function on a neighborhood of $\{0\}\times \{z:|z|\le 1\}\subset {\mathbb{R}}^d\times \mathbb{C}$ 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 $\hat{I}¼$ such that the distribution of $(Y_n-\hat{I}¼n)/\sqrt{n}$ 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$\unicode{x2013}$Lévy) central limit theorem. We also prove the 2020 conjecture of Defant that if $\ddot{I}{€}_n\in {\mathfrak{S}}_n$ is a uniformly random permutation, then the distribution of $(\mathrm{des}(s(\ddot{I}{€}_n))+1-(3-e)n)/\sqrt{n}$ converges, as $n\to \infty$, to a normal distribution with variance $2+2e-e^2$.

8 pages