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On the almost sure invariance principle for stationary sequences of Hilbert-valued random variables. (English) Zbl 1229.60037

Berkes, István (ed.) et al., Dependence in probability, analysis and number theory. A volume in memory of Walter Philipp. Most papers based on the presentations at the conference, Graz, Austria, June 17–20, 2009. Heber City, UT: Kendrick Press (ISBN 0-9793183-8-6/pbk). 157-175 (2010).
The authors prove an almost sure invariance principle for partial sums of a strictly stationary sequence of Hilbert space valued random variables. They obtain as a consequence an extension of the almost sure invariance principle of Dehling and Philipp (1982) for strongly mixing sequences. Applications are given to a Cramer-von Mises statistics, to \(\tau\)-dependent sequences and to a class of Hilbert space valued Markov chains which arises in connection with a class of randomly forced partial differential equations. This is an example of not strongly mixing sequences.
For the entire collection see [Zbl 1196.60010].

MSC:

60F17 Functional limit theorems; invariance principles
60G10 Stationary stochastic processes
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