Central limits from generating functions
Mitchell Lee
TL;DR
The paper establishes a unifying central limit theorem for a sequence of $\mathbb{R}^d$-valued random variables via meromorphic continuation of the reciprocal generating function $f(x,z)^{-1}$. By decomposing $f$ near its pole and expanding the logarithm of the pole location, it derives explicit drift $\mu$ and covariance $\Sigma$ such that $\frac{Y_n-\mu n}{\sqrt{n}} \Rightarrow \mathcal{N}(0,\Sigma)$. It recovers the classical Lindeberg–Lévy CLT for i.i.d. sums and solves Defant's 2020 conjecture on the descent statistic of West's stack-sorting map, obtaining $\mu=3-e$ and $\Sigma=2+2e-e^2$. The method bridges analytic combinatorics and probability by turning analytic properties of generating functions into distributional limits with explicit parameters.
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 $μ$ such that the distribution of $(Y_n - μn) / \sqrt{n}$ 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 $π_n \in \mathfrak{S}_n$ is a uniformly random permutation, then the distribution of $(\operatorname{des} (s(π_n)) + 1 - (3 - e) n) / \sqrt{n}$ converges, as $n \to \infty$, to a normal distribution with variance $2 + 2e - e^2$.