On some variations of the Elephant Random Walk
Le Jeudi 16/11/2023
Salle de séminaire du département de Mathématiques de la FSSM
10h00- 11h00 : Pr. Mohamed El Machkouri
Université de Rouen Normandie, France
On some variations of the Elephant Random Walk
Abstract : The Elephant Random Walk (ERW) was introduced by Schütz and Trimper in 2004 with a view to study memory effects in a one-dimensional discrete-time nearest-neighbor walk on Z with a complete memory of its whole past. The name of the model is inspired by the traditional saying that elephants can always remember anywhere they have been. The memory of the walker is measured in terms of a parameter p between zero and one and the modelexhibits three regimes: diffusive regime (0 < p < 3/4 ), critical regime (p = 3/4 ) and
superdiffusive regime (3/4 < p < 1). The ERW has drawn a lot of attention in the last years and several theoretical results (law of large numbers, central limit theorem, law of the iterated logarithm,…) have been established for each of the three regimes. In 2022, Gut and Stadtmüller introduced an extension of the ERW model allowing the memory of the walker to gradually increase in time. For this new model, we establish central limit theorems in the three regimes and we show how to estimate the memory parameter p of the model. Finaly, we introduce. another new variation of the ERW as a rule for the sequential allocation of drugs in a toxicity-
response study.
