Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

CLT for the zeros of classical random trigonometric polynomials

Jean-Marc Azaïs, Federico Dalmao, and José R. León

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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 show that the variance is equivalent to $V^{2}K\pi$, $0<V^{2}<\infty$, as $K\to\infty$. This last result was recently proved by Su and Shao in (Sci. China Math. 55 (2012) 2347–2366). 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.

Résumé

Nous montrons un Théorème de la Limite Central pour le nombre de racines d’un polynôme trigonométrique aléatoire de la forme $K^{-1/2}\sum_{n=1}^{K}a_{n}\cos(nt)$, ici les $a_{n}$ sont des variables aléatoires Gaussiennes standard et indépendantes. En particulier, nos démontrons que la variance asymptotique du nombre de racines est équivalent à $V^{2}K\pi$, pour une certaine constante $V>0$, lorsque $K\to\infty$. Ce dernier résultat a été récemment démontré par Su and Shao dans (Sci. China Math. 55 (2012) 2347–2366). Notre approche utilise la décomposition dans le chaos d’Itô–Wiener d’une fonctionnelle non linéaire de carré intégrable et la formule de Rice.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 2 (2016), 804-820.

Dates
Received: 23 January 2014
Revised: 6 October 2014
Accepted: 7 October 2014
First available in Project Euclid: 4 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1462367894

Digital Object Identifier
doi:10.1214/14-AIHP653

Mathematical Reviews number (MathSciNet)
MR3498010

Zentralblatt MATH identifier
1342.60025

Subjects
Primary: 60G15: Gaussian processes

Keywords
Classical trigonometric polynomials Random cosines polynomials Number of zeroes CLT Wiener Chaos

Citation

Azaïs, Jean-Marc; Dalmao, Federico; León, José R. CLT for the zeros of classical random trigonometric polynomials. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 804--820. doi:10.1214/14-AIHP653. https://projecteuclid.org/euclid.aihp/1462367894


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