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Article Dans Une Revue Probability Theory and Related Fields Année : 2019

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

Résumé

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 $$ \lim_{n\to\infty}\frac{\text{Var}\left(\mathcal{N}_n([0,\pi],Y)\right)}{n} =\lim_{n\to\infty}\frac{\text{Var}\left(\mathcal{N}_n([0,\pi],G)\right)}{n} +\frac{1}{30}\left(\mathbb{E}(Y_{1,1}^4)-3\right). $$ 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.

Dates et versions

hal-01634848 , version 1 (14-11-2017)

Identifiants

Citer

Vlad Bally, Lucia Caramellino, Guillaume Poly. Non universality for the variance of the number of real roots of random trigonometric polynomials. Probability Theory and Related Fields, 2019, 174 (3-4), pp.887-927. ⟨10.1007/s00440-018-0869-2⟩. ⟨hal-01634848⟩
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