Strong limit theorems for step-reinforced random walks

Zhishui Hu and Yiting Zhang

Stochastic Processes and their Applications, 2024, vol. 178, issue C

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.

Keywords: Reinforcement; Random walk; Strong invariance principles; Martingale (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spa.2024.104484

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