Limit theorems under the Maxwell-Woodroofe condition in Banach spaces

TL;DR

This paper establishes that the Maxwell-Woodroofe condition ensures the law of the iterated logarithm, strong law of large numbers, and weak invariance principle for Banach space-valued stationary processes, extending classical results.

Contribution

It proves the sufficiency and optimality of the Maxwell-Woodroofe condition for key probabilistic limit theorems in Banach spaces, including new maximal inequalities and martingale approximations.

Findings

Maxwell-Woodroofe condition guarantees the law of the iterated logarithm.

The condition is sufficient for the strong law of large numbers in Banach spaces.

The weak invariance principle holds under a version of this condition.

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.