Expected number of real zeros of random Taylor Series

2017/09/09 by Hendrik Flasche, Flasche, Hendrik, Zakhar Kabluchko +1 · 1 citation
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1709.02937

openalex publication_date 2017/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ξ₀,ξ₁,… be i.i.d. random variables with zero mean and unit variance. Consider a random Taylor series of the form f(z)=∑_k=0^∞ ξ_k c_k z^k, where c₀,c₁,… is a real sequence such that c_n² is regularly varying with index γ-1, where γ>0. We prove that 𝔼 N[0,1-ε] ∼ (√γ)/(2π) |log ε| as ε\downarrow 0, where N[0,r] denotes the number of real zeroes of f in the interval [0,r].

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