Mohamed El Machkouri - On some variations of the Elephant Random Walk

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 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 model, we establish central limit
theorems in the three regimes extending previous results by Bercu
(2018), Kubota and Takei (2019) and Gut and Stadtmüller (2022). Finally,
we introduce and investigate the asymptotic properties of another new
variation of the ERW as a rule for the sequential allocation of drugs in
a toxicity-response study.