Elephant random walks; a review

Abstract. In the simple random walk the steps are independent, whereas in the elephant random walk (ERW), which was introduced by Schütz and Trimper in 2004 [33], the next step always depends on the whole path so far. In a series of earlier papers we have investigated some variations and extensions, in particular cases when the elephant has a restricted memory. In the present paper we summarize and extend some results on elephant random walks, closing with some remarks and open questions.

Key words and phrases. Elephant random walks, number of zeros, law of large numbers, central limit theorem, Markov chain.