Strong limit theorems for step-reinforced random walks
Abstract
A step-reinforced random walk is a discrete-time non-Markovian 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 invariance principle 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.
Cite
@article{arxiv.2311.15263,
title = {Strong limit theorems for step-reinforced random walks},
author = {Zhishui Hu and Yiting Zhang},
journal= {arXiv preprint arXiv:2311.15263},
year = {2023}
}