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Strong limit theorems for step-reinforced random walks

Author

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  • Hu, Zhishui
  • Zhang, Yiting

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.

Suggested Citation

  • Hu, Zhishui & Zhang, Yiting, 2024. "Strong limit theorems for step-reinforced random walks," Stochastic Processes and their Applications, Elsevier, vol. 178(C).
    • Handle: RePEc:eee:spapps:v:178:y:2024:i:c:s030441492400190x
    DOI: 10.1016/j.spa.2024.104484
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