Extreme value statistics of positive recurrent centrally biased random walks

TL;DR

This paper analyzes the extreme value behavior of centrally-biased random walks with near-zero drift, revealing how the maximum's distribution relates to the stationary distribution and time scaling in ergodic regimes.

Contribution

It provides a complete characterization of the asymptotic distribution of the maximum for non-translationally invariant Markov chains with zero-drift bias.

Findings

Derived the asymptotic distribution of the maximum

Linked maximum scaling to stationary distribution properties

Enhanced understanding of extreme values in biased random walks

Abstract

We consider the extreme value statistics of centrally-biased random walks with asymptotically-zero drift in the ergodic regime. We fully characterize the asymptotic distribution of the maximum for this class of Markov chains lacking translational invariance, with a particular emphasis on the relation between the time scaling of the expected value of the maximum and the stationary distribution of the process.