A multiple stochastic integral criterion for almost sure limit theorems

Bernard Bercu (1, 2) , Ivan Nourdin (3) , Murad S. Taqqu (4)

1 IMB - Institut de Mathématiques de Bordeaux
2 CQFD - Quality control and dynamic reliability
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
4 Department of Mathematics and Statistics [Boston]

In this paper, we study almost sure central limit theorems for multiple stochastic integrals and provide a criterion based on the kernel of these multiple integrals. We apply our result to normalized partial sums of Hermite polynomials of increments of fractional Brownian motion. We obtain almost sure central limit theorems for these normalized sums when they converge in law to a normal distribution.

Mots clés: Almost sure limit theorem; multiple stochastic integrals; fractional Brownian motion; Hermite power variation

Domaines: Probabilités [math.PR]

Bernard Bercu, Ivan Nourdin, Murad S. Taqqu. A multiple stochastic integral criterion for almost sure limit theorems. 2009. ⟨hal-00375290v1⟩