TY - JOUR

T1 - Expected Number of Zeros of Random Power Series with Finitely Dependent Gaussian Coefficients

AU - Noda, Kohei

AU - Shirai, Tomoyuki

N1 - Funding Information:
The authors are grateful to Professor Akimichi Takemura for a discussion on positive-definiteness of the covariance kernel of the MA(2) model and pointing out the reference []. They would also like to thank the anonymous referee for the careful reading of the manuscript. This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI [JP18H01124] and partially supported by KAKENHI [JP16H06338, JP20H00119, JP20K20884 to T.S.], and K.N. was supported by WISE program (MEXT) at Kyushu University.
Publisher Copyright:
© 2022, The Author(s).

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

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U2 - 10.1007/s10959-022-01203-y

DO - 10.1007/s10959-022-01203-y

M3 - Article

AN - SCOPUS:85142384146

SN - 0894-9840

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

ER -