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.

arXiv commentWe corrected a typo in the definition of the ERW

References in corpus (11)

• Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk
• Elephant Random Walks and their connection to Pólya-type urns
• A martingale approach for the elephant random walk
• Central Limit Theorem for the Elephant Random Walk
• A strong invariance principle for the elephant random walk
• Gaussian fluctuation for superdiffusive elephant random walks
• The Elephant Quantum Walk
• On a class of random walks with reinforced memory
• Limit theorems for a minimal random walk model
• Multi-Dimensional Elephant Random Walk with Coupled Memory
• Finite-Memory Elephant Random Walk and the Central Limit Theorem for Additive Functionals