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

Abstract

AbstractGiven a random walk $$(S_n)$$ ( S n ) with typical step distributed according to some fixed law and a fixed parameter $$p \in (0,1)$$ p ∈ ( 0 , 1 ) , the associated positively step-reinforced random walk is a discrete-time process which performs at each step, with probability $$1-p$$ 1 - p , the same step as $$(S_n)$$ ( 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 $$p < 1/2$$ 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, $$\begin{aligned} \left( B(t) , \, t^p \int _0^t s^{-p} \mathrm {d}B^r(s) , \, t^{-p} \int _0^t s^{p} \mathrm {d}B^c(s) \right) _{t \in \mathbb {R}^+} \end{aligned}$$ B ( t ) , t p ∫ 0 t s - p d B r ( s ) , t - p ∫ 0 t s p d B c ( s ) t ∈ R + for properly correlated Brownian motions $$B, B^r$$ B , B r , $$B^c$$ 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 $$p \in (0,1)$$ p ∈ ( 0 , 1 ) the limiting behaviour of step reinforced random walks. Our method exhausts a martingale approach in conjunction with the martingale functional CLT.