Strong limit theorems for step-reinforced random walks

Abstract

A step-reinforced random walk is a discrete-time process with long range memory. At each step, with a fixed probability $p$, the positively step-reinforced random walk repeats one of its preceding steps chosen uniformly at random, and with complementary probability $1-p$, it has an independent increment. The negatively step-reinforced random walk follows the same reinforcement algorithm but when a step is repeated its sign is also changed. Strong laws of large numbers and strong invariance principles are established for positively and negatively step-reinforced random walks in this work. Our approach relies on two general theorems on the invariance principles for martingale difference sequences and a truncation argument. As by-products of our main results, the law of iterated logarithm and the functional central limit theorem are also obtained for step-reinforced random walks.