Non-Markovian random walks with memory lapses

Abstract

We propose an approach to construct Bernoulli trials {Xi, i ≥ 1} combining dependence and independence periods, and we call it the Bernoulli sequence with random dependence (BSRD). The structure of dependence, in the past Si = X1 + ⋯ + Xi, defines a class of non-Markovian random walks of recent interest in the literature. In this paper, the dependence is activated by an auxiliary collection of Bernoulli trials {Yi, i ≥ 1}, called memory switch sequence. We introduce the concept of memory lapse property, which is characterized by intervals of consecutive independent steps in BSRD. The main results include classical limit theorems for a class of linear BSRD. In particular, we obtain a central limit theorem for a class of BSRD which generalizes some previous results in the literature. Along the paper, several examples of potential applications are provided.