Asymptotic entropy of transformed random walks

TL;DR

This paper extends Abramov's formula to random walks on groups, showing that the asymptotic entropy and rate of escape of transformed walks scale with the expected stopping time, generalizing previous special cases.

Contribution

It proves a general formula relating the asymptotic entropy and escape rate of transformed random walks to the original, using Markov stopping times.

Findings

Asymptotic entropy of transformed walks equals original entropy times expected stopping time.

Rate of escape of transformed walks equals original rate times expected stopping time.

Generalizes Abramov's formula to random walks on groups.

Abstract

We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (resp., rate of escape) of the transformed random walks is equal to the asymptotic entropy (resp., rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov's formula from ergodic theory, its particular cases were established earlier by Kaimanovich [1983] and Hartman, Lima, Tamuz [2014].