Limit laws of the coefficients of polynomials with only unit roots

TL;DR

This paper investigates the asymptotic behavior of random variables with generating functions as polynomials with roots on the unit circle, revealing conditions for normality and describing all possible limit laws.

Contribution

It provides a new criterion for asymptotic normality based on the fourth moment and offers a comprehensive representation theorem for all limit laws of such polynomials.

Findings

Random variables are asymptotically normal if the fourth normalized moment tends to 3.

A representation theorem characterizes all possible limit laws.

Applications span combinatorics, numerical analysis, probability, and algorithms.

Abstract

We consider sequences of random variables whose probability generating functions are polynomials all of whose roots lie on the unit circle. The distribution of such random variables has only been sporadically studied in the literature. We show that the random variables are asymptotically normally distributed if and only if the fourth normalized (by the standard deviation) central moment tends to 3, in contrast to the common scenario for polynomials with only real roots for which a central limit theorem holds if and only if the variance goes unbounded. We also derive a representation theorem for all possible limit laws and apply our results to many concrete examples in the literature, ranging from combinatorial structures to numerical analysis, and from probability to analysis of algorithms.