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