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February 2015 A compact LIL for martingales in $2$-smooth Banach spaces with applications
Christophe Cuny
Bernoulli 21(1): 374-400 (February 2015). DOI: 10.3150/13-BEJ571

Abstract

We prove the compact law of the iterated logarithm for stationary and ergodic differences of (reverse or not) martingales taking values in a separable $2$-smooth Banach space (for instance a Hilbert space). Then, in the martingale case, the almost sure invariance principle is derived from a result of Berger. From those results, we deduce the almost sure invariance principle for stationary processes under the Hannan condition and the compact law of the iterated logarithm for stationary processes arising from non-invertible dynamical systems. Those results for stationary processes are new, even in the real valued case. We also obtain the Marcinkiewicz–Zygmund strong law of large numbers for stationary processes with values in some smooth Banach spaces. Applications to several situations are given.

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Christophe Cuny. "A compact LIL for martingales in $2$-smooth Banach spaces with applications." Bernoulli 21 (1) 374 - 400, February 2015. https://doi.org/10.3150/13-BEJ571

Information

Published: February 2015
First available in Project Euclid: 17 March 2015

zbMATH: 1321.60057
MathSciNet: MR3322323
Digital Object Identifier: 10.3150/13-BEJ571

Rights: Copyright © 2015 Bernoulli Society for Mathematical Statistics and Probability

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Vol.21 • No. 1 • February 2015
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