Joint invariance principles for random walks with positively and negatively reinforced steps

TL;DR

This paper establishes Gaussian limit theorems for positively and negatively reinforced random walks with centered steps, using martingale methods, revealing their asymptotic behavior under certain reinforcement probabilities.

Contribution

It provides the first functional limit theorems for coupled positive and negative reinforced random walks with centered steps, expanding understanding of their asymptotic Gaussian behavior.

Findings

Limiting processes are Gaussian and can be represented via stochastic integrals.

Results hold for reinforcement probability p<1/2 and centered steps.

Method combines martingale approach with functional CLT.

Abstract

Given a random walk $(S_n)$ with typical step distributed according to some fixed law and a fixed parameter $p \in (0,1)$, the associated positively step-reinforced random walk is a discrete-time process which performs at each step, with probability $1-p$, the same step as $(S_n)$ while with probability $p$, it repeats one of the steps it performed previously chosen uniformly at random. The negatively step-reinforced random walk follows the same dynamics but when a step is repeated its sign is also changed. In this work, we shall prove functional limit theorems for the triplet of a random walk, coupled with its positive and negative reinforced versions when $p<1/2$ and when the typical step is centred. As our work will show, the limiting process is Gaussian and admits a simple representation in terms of stochastic integrals. Our method exhausts a martingale approach in conjunction with the martingale functional CLT.