A maximal 𝕃_{𝕡}-inequality for stationary sequences and its applications

Abstract

The paper aims to establish a new sharp Burkholder-type maximal inequality in L p \mathbb {L}_p for a class of stationary sequences that includes martingale sequences, mixingales and other dependent structures. The case when the variables are bounded is also addressed, leading to an exponential inequality for a maximum of partial sums. As an application we present an invariance principle for partial sums of certain maps of Bernoulli shifts processes.