Mohamed El Machkouri – Mathematics and Decision

$\begin{quote} \begin{center} \textbf{On some variations of the Elephant Random Walk} \end{center} \medskip 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 $\mathbb{Z}$ 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 model exhibits 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 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. \end{quote}$