Real Zeros of Random Sums with I.I.D. Coefficients

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 the expected number of real zeros of $P_n$. As a corollary, taking the random variables $\{\eta_k\}$ to be i.i.d.~standard Gaussian, appealing to Fourier inversion we recover the representation for the density function previously given by Vanderbei through 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 $a>0$ and $q\geq 1$, as well as that $\phi$ is three times differentiable with each the second and third derivatives being uniformly bounded, we achieve an upper bound on the density function $\rho_n$ with explicit constants that depend only on the restrictions on $\phi$. As an application we considered the limiting value of $\rho_n$ when the spanning functions $f_k(z)=p_k(z)$, $k=0,1,\dots, n$, where $\{p_k\}$ are Bergman polynomials on the unit disk.