Schütz and Trimper (Phys. Rev. E 70, 045101 (2004)) introduced the ‘elephant random walk (ERW)’ which, like the simple random walk studied over the last century has increments \(+1\) and \(-1\) , however the ERW keeps a track of its past steps. Harbola et al. (Phys. Rev. E, 90, 022136 (2014)) studied a unidirectional ERW where the increments are 0 and \(+1\) . Gut and Stadmüller (J. Appl. Probab. 58, 805–829 (2021), Stat. Probab. Lett. 189, 109598 (2022)) extended the study of the ERW ‘when the elephant has only a restricted memory’ and Laulin (Electron. Commun. Probab., 27, Paper No. 54 (2022)) studied an ERW model with a ‘smooth amnesia’. Here we discuss these models of restricted memory. We also provide a new proof of a result of Miyazaki and Takei (J. Stat. Phys., 181, 587–602 (2020)). Finally we end with some problems connected with these models.
