Expected number of zeros of random power series with finitely dependent Gaussian coefficients

TL;DR

This paper investigates the expected zeros of random power series with Gaussian coefficients, showing they are fewer than in the i.i.d. case and deriving precise asymptotics for finitely dependent coefficients.

Contribution

It provides new bounds and asymptotic formulas for the expected number of zeros in random power series with finitely dependent Gaussian coefficients.

Findings

Expected zeros are fewer than in the hyperbolic GAF with i.i.d. coefficients.

Derived precise asymptotics for zeros when coefficients are finitely dependent.

Negative contributions to zeros relate to zeros of the spectral density.

Abstract

We are concerned with zeros of random power series with coefficients being a stationary, centered, complex Gaussian process. We show that the expected number of zeros in every smooth domain in the disk of convergence is less than that of the hyperbolic GAF with i.i.d. coefficients. When coefficients are finitely dependent, i.e., the spectral density is a trigonometric polynomial, we derive precise asymptotics of the expected number of zeros inside the disk of radius $r$ centered at the origin as $r$ tends to the radius of convergence, in the proof of which we clarify that the negative contribution to the number of zeros stems from the zeros of the spectral density.