Gaussian fluctuations of the elephant random walk with gradually increasing memory

The elephant random walk (ERW) is a discrete-time random walk introduced by Schutz and Trimper (2004) in order to investigate how long-range memory affects the behavior of the random walk. Its particularity is that the next step of the walker depends on its whole past through a parameter p is an element of[0,1] . In this work, we investigate the validity of the central limit theorem of the ERW when the walker has only a gradually increasing memory. Our contribution provides a positive answer to a conjecture raised in a recent work by Gut and Stadtmuller (2022 Stat. Probab. Lett. 189 109598).