Zeros of sections of power series: deterministic and random

TL;DR

This paper provides a simplified proof of a classical characterization of Szeg"o's power series and applies it to analyze the asymptotic distribution of zeros of sections of both deterministic and random power series, including new results.

Contribution

It offers a streamlined proof of a known characterization and extends the analysis to the asymptotic zero distribution of random power series sections.

Findings

New results on zero distribution of random power series

Refined understanding of deterministic zero behavior

Connections between deterministic and random zero distributions

Abstract

We present a streamlined proof (and some refinements) of a characterization (due to F. Carlson and G. Bourion, and also to P. Erd\"os and H. Fried) of the so called Szeg\"o's power series. This characterization is then applied to readily obtain some (more) recent known results and some new results on the asymptotic distribution of zeros of sections of random power series, extricating quite naturally the deterministic ingredients. Finally, we study the possible limits of the zero counting probabilities of a power series.