Elephant Random Walk with a Random Step Size and Gradually Increasing Memory and Delays

Abstract

The ERW model was introduced twenty years ago to study memory effects in a one-dimensional discrete-time random walk with a complete memory of its past throughout a parameter p between zero and one. Several variations of the ERW model have recently been introduced. In this work, we investigate the asymptotic normality of the ERW model with a random step size and gradually increasing memory and delays. In particular, we extend some recent results in this subject.

Keywords

• elephant random walk
• central limit theorem
• asymptotic distributions for expressions
• normal distribution
• Mittag-Leffler distribution
• phase transition