Non universality for the variance of the number of real roots of random trigonometric polynomials

Abstract

In this article, we consider the following family of random trigonometric polynomials $p_n(t,Y)={\sum }_{k=1}^n{Y}_{k,1}\cos (kt)+Y_{k,2}\sin (kt)$ for a given sequence of i.i.d. random variables $\{Y_{k,1},Y_{k,2}{\}}_{k\ge 1}$ which are centered and standardized. We set $\mathcal{N}([0,\pi ],Y)$ the number of real roots over $[0,\pi ]$ and $\mathcal{N}([0,\pi ],G)$ the corresponding quantity when the coefficients follow a standard Gaussian distribution. We prove under a Doeblin's condition on the distribution of the coefficients that $$\underset{n\to \infty }{\lim }\frac{\text{Var}\left({\mathcal{N}}_n([0,\pi ],Y)\right)}{n}=\underset{n\to \infty }{\lim }\frac{\text{Var}\left({\mathcal{N}}_n([0,\pi ],G)\right)}{n}+\frac{1}{30}\left(\mathbb{E}(Y_{1,1}^4)-3\right).$$ The latter establishes that the behavior of the variance is not universal and depends on the distribution of the underlying coefficients through their kurtosis. Actually, a more general result is proven in this article, which does not require that the coefficients are identically distributed. The proof mixes a recent result regarding Edgeworth's expansions for distribution norms established in arXiv:1606.01629 with the celebrated Kac-Rice formula.

References in corpus (1)

• Expected number of real zeros of random Taylor Series

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