Expected Number of Zeros of Random Power Series with Finitely Dependent Gaussian Coefficients

Kohei Noda () and Tomoyuki Shirai ()
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Kohei Noda: Kyushu University
Tomoyuki Shirai: Kyushu University

Journal of Theoretical Probability, 2023, vol. 36, issue 3, 1534-1554

Abstract: Abstract We are concerned with zeros of random power series with coefficients being a stationary, centered, complex Gaussian process. We show that the expected number of zeros in every smooth domain in the disk of convergence is less than that of the hyperbolic Gaussian analytic function with i.i.d. coefficients. When coefficients are finitely dependent, i.e., the spectral density is a trigonometric polynomial, we derive precise asymptotics of the expected number of zeros inside the disk of radius r centered at the origin as r tends to the radius of convergence, in the proof of which we clarify that the negative contribution to the number of zeros stems from the zeros of the spectral density.

Keywords: Gaussian analytic function; Stationary Gaussian process; Point process; Primary: 30B20; Secondary: 60G15; 60G55 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01203-y

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