Finite-Memory Elephant Random Walk and the Central Limit Theorem for Additive Functionals

2019/11/13 by Iddo Ben-Ari, Ben-Ari, Iddo, Jonah Green +7 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #Gene Regulatory Network Analysis #Stochastic processes and statistical mechanics #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.1911.05716

Abstract

The Central Limit Theorem (CLT) for additive functionals of Markov chains is a well known result with a long history. In this paper we present applications to two finite-memory versions of the Elephant Random Walk, solving a problem from arXiv:1812.01915. We also present a derivation of the CLT for additive functionals of finite state Markov chains, which is based on positive recurrence, the CLT for IID sequences and some elementary linear algebra, and which focuses on characterization of the variance.

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• Limit theorems for elephant random walks remembering the very recent past, with applications to the Takagi-van der Waerden class functions

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