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

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  • Kohei Noda

    (Kyushu University)

  • Tomoyuki Shirai

    (Kyushu University)

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.

Suggested Citation

  • Kohei Noda & Tomoyuki Shirai, 2023. "Expected Number of Zeros of Random Power Series with Finitely Dependent Gaussian Coefficients," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1534-1554, September.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:3:d:10.1007_s10959-022-01203-y
    DOI: 10.1007/s10959-022-01203-y
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    References listed on IDEAS

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    1. T. W. Anderson & Akimichi Takemura, 1986. "Why Do Noninvertible Estimated Moving Averages Occur?," Journal of Time Series Analysis, Wiley Blackwell, vol. 7(4), pages 235-254, July.
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