Limit theorems under the Maxwell-Woodroofe condition in Banach spaces

Abstract

We prove that, for (adapted) stationary processes, the so-called Maxwell-Wood-roofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. We obtain a similar conclusion concerning the Marcinkiewicz-zygmund strong law of large numbers. Those results actually hold in the context of Banach valued stationary processes, including the case of $L^r$-valued random variables, with $1\le r<\infty$. In this setting we also prove the weak invariance principle, under a version of the Maxwell-Woodroofe condition, generalizing a result of Peligrad and Utev \cite{PU}. Our results extend to non-adapted processes as well, and, partly to stationary processes arising from dynamical systems. The proofs make use of a new maximal inequality and of approximation by martingales, for which some of our results are also new.

References in corpus (5)

• Martingale approximations for sums of stationary processes
• On martingale approximations and the quenched weak invariance principle
• Strong invariance principles with rate for "reverse" martingales and applications
• Independence of Four Projective Criteria for the Weak Invariance Principle
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Cited by in corpus (1)

• H{ö}lderian weak invariance principle under Maxwell and Woodroofe condition