On a class of random walks with reinforced memory

Abstract

This paper deals with different models of random walks with a reinforced memory of preferential attachment type. We consider extensions of the Elephant Random Walk introduced by Schütz and Trimper [2004] with a stronger reinforcement mechanism, where, roughly speaking, a step from the past is remembered proportional to some weight and then repeated with probability $p$. With probability $1-p$, the random walk performs a step independent of the past. The weight of the remembered step is increased by an additive factor $b\ge 0$, making it likelier to repeat the step again in the future. A combination of techniques from the theory of urns, branching processes and $\alpha$-stable processes enables us to discuss the limit behavior of reinforced versions of both the Elephant Random Walk and its $\alpha$-stable counterpart, the so-called Shark Random Swim introduced by Businger [2018]. We establish phase transitions, separating subcritical from supercritical regimes.

arXiv comment38 pages, 1 figure

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Cited by in corpus (7)

• Functional limit theorems for the Multi-dimensional Elephant Random Walk
• Joint invariance principles for random walks with positively and negatively reinforced steps
• Multidimensional walks with random tendency
• Asymptotic Normality of Superdiffusive Step-Reinforced Random Walks
• Broadcasting induced colourings of random recursive trees and preferential attachment trees
• Limit theorems for a random walk with memory perturbed by a dynamical system
• Ulam's History-dependent Random Adding Process