CLT for the zeros of Classical Random Trigonometric Polynomials

Abstract

We prove a Central Limit Theorem for the number of zeros of random trigonometric polynomials of the form $K^{-1/2}{\sum }_{n=1}^K{a}_n\cos (nt)$, being $(a_n{)}_n$ independent standard Gaussian random variables. In particular, we prove the conjecture by Farahmand, Granville & Wigman that the variance is equivalent to $V^{2}K$, $0<V^2<\infty$, as $K\to \infty$. % The case of stationary trigonometric polynomials was studied by Granville & Wigman and by Aza\"\is & León. Our approach is based on the Hermite/Wiener-Chaos decomposition for square-integrable functionals of a Gaussian process and on Rice Formula for zero counting.

27 pages. To appear in L'Annales de l'Institut Henri Poincaré, Serie B

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