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Local universality of the number of zeros of random trigonometric polynomials with continuous coefficients

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Abstract

Let $X_N$ be a random trigonometric polynomial of degree $N$ with iid coefficients and let $Z_N(I)$ denote the (random) number of its zeros lying in the compact interval $I\subset\mathbb{R}$. Recently, a number of important advances were made in the understanding of the asymptotic behaviour of $Z_N(I)$ as $N\to\infty$, in the case of standard Gaussian coefficients. The main theorem of the present paper is a universality result, that states that the limit of $Z_N(I)$ does not really depend on the exact distribution of the coefficients of $X_N$. More precisely, assuming that these latter are iid with mean zero and unit variance and have a density satisfying certain conditions, we show that $Z_N(I)$ converges in distribution toward $Z(I)$, the number of zeros within $I$ of the centered stationary Gaussian process admitting the cardinal sine for covariance function.

Dates and versions

hal-01254543 , version 1 (12-01-2016)

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Jean-Marc Azaïs, Federico Dalmao, José León, Ivan Nourdin, Guillaume Poly. Local universality of the number of zeros of random trigonometric polynomials with continuous coefficients. 2015. ⟨hal-01254543⟩
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