Moments of recurrence times for Markov chains

TL;DR

This paper generalizes classical results on moments of return times in Markov chains, identifying the precise conditions on functions under which finiteness of moments for one state implies finiteness for all states.

Contribution

It extends classical Markov chain recurrence results to a broader class of functions, characterizing when moments of return times are uniformly finite across states.

Findings

Functions that do not grow exponentially satisfy the generalized recurrence moment property.

The classical case with power functions $f(n)=n^p$ is a special instance of this broader class.

The paper identifies the best possible conditions on functions for the property to hold.

Abstract

We consider moments of the return times (or first hitting times) in a discrete time discrete space Markov chain. It is classical that the finiteness of the first moment of a return time of one state implies the finiteness of the first moment of the first return time of any other state. We extend this statement to moments with respect to a function $f$, where $f$ satisfies a certain, best possible condition. This generalizes results of K. L. Chung (1954) who considered the functions $f(n)=n^p$ and wondered "[...] what property of the power $n^p$ lies behind this theorem [...]" (see Chung (1967), p. 70). We exhibit that exactly the functions that do not increase exponentially -- neither globally nor locally -- fulfill the above statement.