Non-Markovian random walks with memory lapses

TL;DR

This paper introduces Bernoulli sequences with random dependence (BSRD), modeling non-Markovian random walks with memory lapses, and establishes classical limit theorems including a generalized central limit theorem.

Contribution

It proposes a novel framework for constructing dependent Bernoulli sequences with memory lapses and proves classical limit theorems for this new class.

Findings

Established a central limit theorem for BSRD.

Characterized the memory lapses property in non-Markovian walks.

Provided examples illustrating potential applications.

Abstract

We propose an approach to construct Bernoulli trials $\{X_i, i\ge 1\}$ combining dependence and independence periods, and call it Bernoulli sequence with random dependence (BSRD). The structure of dependence, on the past $S_i = X_1 + \ldots + X_i$, {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 $\{Y_i, i\ge 1\}$, called {\it memory switch sequence}. We introduce the concept of {\it memory lapses 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 literature. Along the paper, several examples of potential applications are provided.