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