Bound on the maximal function associated to the law of the iterated logarithms for Bernoulli random fields

TL;DR

This paper establishes a sufficient condition for the bounded law of the iterated logarithms in stationary Bernoulli random fields, using maximal function moment control, with applications to various stochastic processes.

Contribution

It introduces a new sufficient condition for bounded LIL in Bernoulli random fields based on maximal function moments, extending previous results to broader classes of processes.

Findings

Established a sufficient condition for bounded LIL in Bernoulli random fields.

Applied the condition to linear, Gaussian, and Volterra processes.

Demonstrated the effectiveness of the approach through multiple applications.

Abstract

We provide a sufficient condition for the bounded law of the iterated logarithms for strictly stationary random fields expressable as a functional of i.i.d. random fields when the summation is done on rectangles. The study is done via the control of the moments of an appropriated maximal function. Applications to functionals of linear random fields, functions of a Gaussian linear random field and Volterra process are given.