Joint Invariance Principles for Random Walks with Positively and Negatively Reinforced Steps.

Given a random walk (S n) with typical step distributed according to some fixed law and a fixed parameter p ∈ (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. The limiting process is Gaussian and admits a simple representation in terms of stochastic integrals, B (t) , t p ∫ 0 t s - p d B r (s) , t - p ∫ 0 t s p d B c (s) t ∈ R + <graphic href="10955_2022_2993_Article_Equ45.gif"></graphic> for properly correlated Brownian motions B , B r , B c . The processes in the second and third coordinate are called the noise reinforced Brownian motion (as named in [1]), and the noise counterbalanced Brownian motion of B. Different couplings are also considered, allowing us in some cases to drop the centredness hypothesis and to completely identify for all regimes p ∈ (0 , 1) the limiting behaviour of step reinforced random walks. Our method exhausts a martingale approach in conjunction with the martingale functional CLT. [ABSTRACT FROM AUTHOR]