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Almost sure limit theorems on Wiener chaos: the non-central case. (English) Zbl 1412.60041
Summary: In [B. Bercu et al., Stochastic Processes Appl. 120, No. 9, 1607–1628 (2010; Zbl 1219.60020)], 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 [loc. cit.], 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 [loc. cit.].

MSC:
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
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