2024/06/25 by Mitchell Lee, Lee, Mitchell
Computer Science · Decision Sciences · Mathematics · #60F05 #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2406.17874
openalex publication_date 2024/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (Yn)n be a sequence of ℝd-valued random variables. Suppose that the generating function f(x, z) = ∑n = 0^∞ φYn(x) zn, where φYn is the characteristic function of Yn, extends to a function on a neighborhood of \0\ × \z : |z| ≤ 1\ ⊂ ℝd × 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 (Yn - μn) / √(n) converges weakly to a normal distribution as n → ∞. If Yn = X1 + ⋯ + Xn, where (Xn)n are i.i.d. random variables, then we recover the classical (Lindeberg\unicodex2013Lévy) central limit theorem. We also prove the 2020 conjecture of Defant that if πn ∈ \mathfrakSn is a uniformly random permutation, then the distribution of (des (s(πn)) + 1 - (3 - e) n) / √(n) converges, as n → ∞, to a normal distribution with variance 2 + 2e - e2.