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The distribution of the zeros of random trigonometric polynomials
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 133, Number 2, April 2011
- pp. 295-357
- 10.1353/ajm.2011.0015
- Article
- Additional Information
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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.