The distribution of the zeroes of random trigonometric polynomials

Details The distribution of the zeroes of random trigonometric polynomials The distribution of the zeroes of random trigonometric polynomials

2008-09-10

arxiv.org/abs/0809.1848v2

Mathematics

Probability

51 pages. We cut the size of the paper to better suit publication. In particular, all the results of empirical experiments wer

Scientific paper

We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\to\infty$. 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 $\frac{Z_{N}-\E Z_{N}}{\sqrt{cN}}$ converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.

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