The elephant random walk with gradually increasing memory

In the simple random walk steps are independent, viz., walker has no memory. contrast, in elephant (ERW), which was introduced by Schütz and Trimper (2004), next step always depends on whole path so far. Various authors have studied further properties of ERW. Gut Stadtmüller (2021b) we case when Elephant remembers only a finite part first or last steps. both cases there separation into two different regimes as classical We also posed question about what happens if she gradually increasing past. This paper will give some answers to that question. discuss related questions for ERW:s with delays.