References & Citations
Mathematics > Probability
Title: The number of real zeros of elliptic polynomials
(Submitted on 21 Nov 2021 (v1), last revised 8 May 2024 (this version, v4))
Abstract: Let $N_n(a, b)$ denote the number of real zeros of Gaussian elliptic polynomials of degree $n$ on the interval $(a, b)$, where $a$ and $b$ may vary with $n$. We obtain a precise formula for the variance of $N_n(a, b)$ and utilize this expression to derive an asymptotic expansion for large values of $n$. Furthermore, we provide sharp estimates for the cumulants and central moments of $N_n(a, b)$. These estimates are instrumental in establishing sufficient conditions on the interval $(a, b)$ for $N_n(a, b)$ to satisfy both a central limit theorem and a strong law of large numbers. In the second part of the paper, we extend our analysis to nondegenerate Gaussian analytic functions, including well-known examples such as the Gaussian Weyl series and Weyl polynomials.
Submission history
From: Nhan Nguyen [view email][v1] Sun, 21 Nov 2021 18:49:50 GMT (76kb,D)
[v2] Sun, 19 Feb 2023 22:18:51 GMT (245kb,D)
[v3] Sat, 2 Dec 2023 00:46:29 GMT (257kb,D)
[v4] Wed, 8 May 2024 00:03:29 GMT (257kb,D)
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