The Complex Roots of Random Sums

TL;DR

This paper generalizes the study of roots of random polynomials to sums of basis functions, providing explicit density formulas and analyzing their distribution in the complex plane, including practical examples like Fourier series.

Contribution

It introduces a generalized framework for analyzing roots of random sums of basis functions with explicit density formulas and asymptotic behavior.

Findings

Derived explicit density formulas for roots distribution

Analyzed limiting density as the number of terms increases

Included practical examples such as Fourier series

Abstract

This paper extends earlier work on the distribution in the complex plane of the roots of random polynomials. In this paper, the random polynomials are generalized to random finite sums of given "basis" functions. The basis functions are assumed to be entire functions that are real-valued on the real line. The coefficients are assumed to be independent identically distributed Normal $(0,1)$ random variables. An explicit formula for the density function is given in terms of the set of basis functions. We also consider some practical examples including Fourier series. In some cases, we derive an explicit formula for the limiting density as the number of terms in the sum tends to infinity.