Limit theorems for a random walk with memory perturbed by a dynamical system

Abstract

We introduce a new random walk with unbounded memory obtained as a mixture of the elephant random walk and the dynamic random walk, which we call the Dynamic Elephant Random Walk (DERW). As a consequence of this mixture, the distribution of the increments of the resulting random process is time dependent. We prove a strong law of large numbers for the DERW and, in a particular case, we provide an explicit expression for its speed. Finally, we give sufficient conditions for the central limit theorem and the law of the iterated logarithm to hold.