Real zeros of random sums with i.i.d. coefficients
Volume 161 / 2020
Abstract
Let $\{f_k\}$ be a sequence of entire functions that are real valued on the real-line. We study the expected number of real zeros of random sums of the form $P_n(z)=\sum _{k=0}^n\eta _k f_k(z)$, where $\{\eta _k\}$ are real valued i.i.d. random variables. We establish a formula for the density function $\rho _n$ for that number. As a corollary, taking $\{\eta _k\}$ to be i.i.d. standard Gaussian, appealing to Fourier inversion we recover the representation for the density function previously given by Vanderbei by means of a different proof. Placing the restrictions on the common characteristic function $\phi $ of $\{\eta _k\}$ that $|\phi (s)|\leq (1+as^2)^{-q}$, with $a \gt 0$ and $q\geq 1$, as well as that $\phi $ is three times differentiable with both the second and third derivatives uniformly bounded, we achieve an upper bound on $\rho _n$ with explicit constants that depend only on the restrictions on $\phi $. As an application we consider the limiting value of $\rho _n$ when the spanning functions $f_k(z)$ are $p_k(z)$, $k=0,1,\dots , n$, where $\{p_k\}$ are the Bergman polynomials on the unit disk.
