Asymmetric Elephant Random Walk

Sunday, October 27, 2024, 4:15 -- 16:45

Elephant random walk has become a subject of great interest since it was introduced by Schutz and Trimper. It generalizes the simple symmetric random walk, where a step is chosen uniformly from the entire past history of the walk and then the new move is made by repeating the step with certain probability or flipping it with complementary probability. We introduce an asymmetric version of it, where the flip probabilities depend on the sign of the chosen step. By considering a representation in terms of two color urns, we provide SLLN and fluctuations around the SLLN limit. The fluctuations show a phase transition from diffusive to superdiffusive behavior depending on the flip probabilities. The limiting distribution of the fluctuations are Gaussian in subcritical and critical cases. We also provide Berry-Esseen type bounds for the rates of convergence in these two cases.
This is a joint work with Aritra Majumdar.