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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2111.10875 |
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| _version_ | 1866911935289622528 |
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| author | Nguyen, Nhan D. V. |
| author_facet | Nguyen, Nhan D. V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_10875 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The number of real zeros of elliptic polynomials Nguyen, Nhan D. V. Probability 60G15, 60G50, 60F05, 41A60 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. |
| title | The number of real zeros of elliptic polynomials |
| topic | Probability 60G15, 60G50, 60F05, 41A60 |
| url | https://arxiv.org/abs/2111.10875 |