Expected number of real zeros of random Taylor Series

Abstract

Let $ξ_0,ξ_1,\ldots$ be i.i.d. random variables with zero mean and unit variance. Consider a random Taylor series of the form $f(z)=\sum_{k=0}^\infty ξ_k c_k z^k$, where $c_0,c_1,\ldots$ is a real sequence such that $c_n^2$ is regularly varying with index $γ-1$, where $γ>0$. We prove that $\mathbb{E} N[0,1-ε] \sim \frac{\sqrtγ}{2π} |\log ε|$ as $ε\downarrow 0$, where $N[0,r]$ denotes the number of real zeroes of $f$ in the interval $[0,r]$.