Expected number of real zeros of random Taylor Series

Abstract

Let ${\xi }_0,{\xi }_1,\dots$ be i.i.d. random variables with zero mean and unit variance. Consider a random Taylor series of the form $f(z)={\sum }_{k=0}^{\infty }{\xi }_k{c}_k{z}^k$, where $c_0,c_1,\dots$ is a real sequence such that $c_n^2$ is regularly varying with index $\gamma -1$, where $\gamma >0$. We prove that $\mathbb{E}N[0,1-\epsilon ]\sim \frac{\sqrt{\gamma }}{2\pi }|\log \epsilon |$ as $\epsilon \downarrow 0$, where $N[0,r]$ denotes the number of real zeroes of $f$ in the interval $[0,r]$.

36 pages

References in corpus (1)

• Universality of the nodal length of bivariate random trigonometric polynomials

Cited by in corpus (1)

• Non universality for the variance of the number of real roots of random trigonometric polynomials