2019/03/15 by Yeager, Aaron M.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1903.06642
Let \fk\ 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 Pn(z)=∑k=0nηk fk(z), where \ηk\ are real valued i.i.d.~random variables. We establish a formula for the density function ρn for the expected number of real zeros of Pn. As a corollary, taking the random variables \ηk\ 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 ϕ of \ηk\ that |ϕ(s)|≤ (1+as2)-q, with a>0 and q≥ 1, as well as that ϕ is three times differentiable with each the second and third derivatives being uniformly bounded, we achieve an upper bound on the density function ρn with explicit constants that depend only on the restrictions on ϕ. As an application we considered the limiting value of ρn when the spanning functions fk(z)=pk(z), k=0,1,…, n, where \pk\ are Bergman polynomials on the unit disk.