Probabilistic Limit Theorems Induced by the Zeros of Polynomials

TL;DR

This paper investigates sequences of discrete random variables with probability generating functions free of zeros in a complex sector, deriving limit theorems and concentration bounds, and providing new proofs for specific polynomial classes.

Contribution

It introduces sharp cumulant bounds for zero-free generating functions, leading to various probabilistic limit theorems and an improved proof for polynomials with roots on the unit circle.

Findings

Established Berry-Esseen bounds for the distributions.

Derived moderate deviation and concentration inequalities.

Provided an alternative proof with better constants for certain polynomials.

Abstract

Sequences of discrete random variables are studied whose probability generating functions are zero-free in a sector of the complex plane around the positive real axis. Sharp bounds on the cumulants of all orders are stated, leading to Berry-Esseen bounds, moderate deviation results, concentration inequalities and mod-Gaussian convergence. In addition, an alternate proof of the cumulant bound with improved constants for a class of polynomials all of whose roots lie on the unit circle is provided. A variety of examples is discussed in detail.