The density of complex zeros of random sums

TL;DR

This paper derives an exact formula for the expected density of complex zeros of random sums involving Gaussian coefficients and analytic functions, analyzing their behavior as the number of terms grows large.

Contribution

It provides a novel exact formula for the zero density of random sums with Gaussian coefficients and explores their asymptotic behavior and special cases.

Findings

Exact density formula for zeros of random sums

Asymptotic behavior of the density as N increases

Numerical and empirical analysis of specific cases

Abstract

Let $\{\eta_{j}\}_{j = 0}^{N}$ be a sequence of independent, identically distributed random complex Gaussian variables, and let $\{f_{j} (z)\}_{j = 0}^{N}$ be a sequence of given analytic functions that are real-valued on the real number line. We prove an exact formula for the expected density of the distribution of complex zeros of the random equation $\sum_{j = 0}^{N} \eta_{j} f_{j} (z) = \mathbf{K}$, where $\mathbf{K} \in \mathds{C}$. The method of proof employs a formula for the expected absolute value of quadratic forms of Gaussian random variables. We then obtain the limiting behaviour of the density function as $N$ tends to infinity and provide numerical computations for the density function and empirical distributions for random sums with certain functions $f_{j} (z)$. Finally, we study the case when the $f_{j} (z)$ are polynomials orthogonal on the real line and the unit circle.