Electronic Communications in Probability

Almost sure limit theorems on Wiener chaos: the non-central case

Ehsan Azmoodeh and Ivan Nourdin

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In [1], a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-Itô integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in [1], by considering the more general situation where the sequence $(G_n)$ may not need to converge to a Gaussian distribution. As an application, we prove that partial sums of Hermite polynomials of increments of fractional Brownian motion satisfy an almost sure limit theorem in the long-range dependence case, a problem left open in [1].

Article information

Electron. Commun. Probab., Volume 24 (2019), paper no. 9, 12 pp.

Received: 3 October 2018
Accepted: 18 January 2019
First available in Project Euclid: 15 February 2019

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Primary: 60F05: Central limit and other weak theorems 60G22: Fractional processes, including fractional Brownian motion 60G15: Gaussian processes 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus

almost sure limit theorem multiple Wiener-Itô integrals Malliavin calculus characteristic function Wiener chaos Hermite distribution fractional Brownian motion

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Azmoodeh, Ehsan; Nourdin, Ivan. Almost sure limit theorems on Wiener chaos: the non-central case. Electron. Commun. Probab. 24 (2019), paper no. 9, 12 pp. doi:10.1214/19-ECP212. https://projecteuclid.org/euclid.ecp/1550199821

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