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 Gut and Stadtmüeller (2018). 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.
Research performed during Markov Chains REU, partially supported by NSA grant H98230-19-1-0022 to Iddo Ben-Ari. The fourth author was supported at the Technion by a Zuckerman Fellowship.
"Finite-memory elephant random walk and the central limit theorem for additive functionals." Braz. J. Probab. Stat. 35 (2) 242 - 262, May 2021. https://doi.org/10.1214/20-BJPS475