CLT for the zeros of Classical Random Trigonometric Polynomials

Jean-Marc Azaïs (1) , Federico Dalmao (2) , José R. León (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 UDELAR - Universidad de la República [Montevideo]
3 Escuela de Matematicas [Caracas]

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 $V^2K$, $0

Mots clés: Rice Formula; Wiener Chaos expansion; zeros of a random trigonometric polynomial

Domaines: Probabilités [math.PR]

Jean-Marc Azaïs, Federico Dalmao, José R. León. CLT for the zeros of Classical Random Trigonometric Polynomials. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2016, 52 (2), pp.804--820. ⟨10.1214/14-AIHP653⟩. ⟨hal-00942047⟩