A compact LIL for martingales in $2$-smooth Banach spaces with applications

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.

Published at http://dx.doi.org/10.3150/13-BEJ571 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• Strong invariance principles with rate for "reverse" martingales and applications
• A quenched invariance principle for stationary processes
• The almost sure invariance principle for unbounded functions of expanding maps

Cited by in corpus (4)

• Limit theorems for U-statistics of Bernoulli data
• Limit theorems under the Maxwell-Woodroofe condition in Banach spaces
• Bound on the maximal function associated to the bounded law of the iterated logarithms via orthomartingale approximation
• Bound on the maximal function associated to the law of the iterated logarithms for Bernoulli random fields