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The distribution of the zeros of random trigonometric polynomials

Granville, Andrew and Wigman, Igor 2011. The distribution of the zeros of random trigonometric polynomials. American Journal of Mathematics 133 (2) , pp. 295-357. 10.1353/ajm.2011.0015

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We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\rightarrow\infty$. It is known that as $N$ grows to infinity, the expected number of the zeros is asymptotic to $\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that $\frac{Z_{N}-{\Bbb E} Z_{N}}{\sqrt{cN}}$ converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: Johns Hopkins University Press
ISSN: 1080-6377
Last Modified: 26 Jun 2019 01:57

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