Central limit theorems and the geometry of polynomials

2019/08/23 by Marcus Michelen, Julian Sahasrabudhe, Michelen, Marcus +1 · 2 citations
Mathematics · #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Functional Equations Stability Results #Mathematics and Applications #Point processes and geometric inequalities #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1908.09020

openalex publication_date 2019/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X ∈ \0,…,n \ be a random variable, with mean μ and standard deviation σ and let f_X(z) = ∑_k ℙ(X = k) z^k, be its probability generating function. Pemantle conjectured that if σ is large and f_X has no roots close to 1∈ ℂ then X must be approximately normal. We completely resolve this conjecture in the following strong quantitative form, obtaining sharp bounds. If δ= min_ζ|ζ-1| over the complex roots ζ of f_X, and X^∗ := (X-μ)/σ, then sup_{t ∈ ℝ} |ℙ(X^∗ ≤ t) - ℙ( Z ≤ t) | = O((log n)/(δσ) ) where Z ∼ N(0,1) is a standard normal. This gives the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer. We also show that if f_X has no roots with small argument, then X must be approximately normal, again in a sharp quantitative form: if we set δ= min_ζ|arg(ζ)| then sup_{t ∈ ℝ} |ℙ(X^∗ ≤ t) - ℙ( Z ≤ t) | = O((1)/(δσ) ). Using this result, we answer a question of Ghosh, Liggett and Pemantle by proving a sharp multivariate central limit theorem for random variables with real-stable probability generating functions.

Citations

• Four Deviations Suffice for Rank 1 Matrices

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