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Expected number of real zeroes of random Taylor series. (English) Zbl 1451.30016

Let \(\xi_0, \xi_1, \ldots \) be a sequence of independent, identically distributed real-valued random variables with \(\mathbb{E}[\xi_0]=0\) and \(\mathbb{E}[\xi_0^2]=1\). The authors study random Taylor series of the form \(f(z)=\sum_{k=0}^{\infty} \xi_kc_kz^k\) under the following assumption on the sequence of weights: \(c_n^2=\frac{n^{\gamma-1}}{\Gamma(\gamma)}L(n)\), where \(\Gamma\) denotes the Gamma function, \(L\) is slowly varying function, and \(\gamma>0\). This assumption ensures that the radius of convergence of the series is 1 with probability 1, and the expected number of real zeros of the series is infinite. In the paper the authors describe the speed of clustering of real zeros of \(f\) near the point 1. They prove: \[\mathbb{E}N[0,r]\ \sim \ \frac{\sqrt{\gamma}}{2\pi} \log \frac{1}{1-r}, \quad \text{as } r \uparrow 1.\] Here \(N[0, r]\) denotes the number of real zeros of \(f\) in the interval \([0, r]\). In order to obtain this result the authors develop the method of I. A. Ibragimov and N. B. Maslova [Theory Probab. Appl. 16, 228–248 (1971; Zbl 0277.60051); translation from Teor. Veroyatn. Primen. 16, 229–248 (1971)].

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30B20 Random power series in one complex variable
26C10 Real polynomials: location of zeros
60F99 Limit theorems in probability theory
60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
60G15 Gaussian processes

Citations:

Zbl 0277.60051
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References:

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