Reinforced random walks under memory lapses

TL;DR

This paper studies a one-dimensional reinforced random walk with memory lapses, analyzing its long-term behavior, diffusive properties, and establishing limit theorems using martingale methods.

Contribution

It introduces a new model of reinforced random walk with probabilistic memory lapses and provides a comprehensive asymptotic analysis including laws of large numbers and limit theorems.

Findings

Characterization of diffusive and superdiffusive regimes

Proof of law of large numbers for the process

Establishment of central limit theorems and law of iterated logarithm

Abstract

We introduce a one-dimensional random walk, which at each step performs a reinforced dynamics with probability $\theta$ and with probability $1 - \theta$, the random walk performs a step independent of the past. We analyse its asymptotic behaviour, showing a law of large numbers and characterizing the diffusive and superdiffusive regions. We prove central limit theorems and law of iterated logarithm based on the martingale approach.