On a class of unbalanced step-reinforced random walks

TL;DR

This paper introduces a unified framework for unbalanced step-reinforced random walks, encompassing the elephant random walk and both positively and negatively reinforced variants through a memory-parameter $a=(2p-1)\alpha$. Using martingale techniques and a careful truncation strategy, it establishes a strong law of large numbers $\frac{T_n}{n} \to \frac{(1-\alpha)\mu_1}{1-a}$ and a spectrum of central limit theorems that depend on $a$, including diffusive ($a<\tfrac12$), critical ($a=\tfrac12$), and superdiffusive ($a>\tfrac12$) regimes with explicit variances. It also extends the results to a gradually increasing memory setting with memory gap $m_n$, yielding CLTs with a mixed variance that depends on both $a$ and $\theta=\lim m_n/n$, thereby linking standard and triangular-array approaches. The findings provide a cohesive theoretical basis for long-range-dependent random walks with reinforcement and offer precise asymptotics across regimes, with potential applications to models of reinforced processes and related urn schemes.

Abstract

A step-reinforced random walk is a discrete-time stochastic process with long-range dependence. At each step, with a fixed probability $α$, the so-called positively step-reinforced random walk repeats one of its previous steps, chosen randomly and uniformly from its entire history. Alternatively, with probability $1-α$, it makes an independent move. For the so-called negatively step-reinforced random walk, the process is similar, but any repeated step is taken with its direction reversed. These random walks have been introduced respectively by Simon (1955) and Bertoin (2024) and are sometimes refered to the self-confident step-reinforced random walk and the counterbalanced step-reinforced random walk respectively. In this work, we introduce a new class of unbalanced step-reinforced random walks for which we prove the strong law of large numbers and the central limit theorem. In particular, our work provides a unified treatment of the elephant random walk introduced by Schutz and Trimper (2004) and the positively and negatively step-reinforced random walks.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4