The Cover Time of a (Multiple) Markov Chain with Rational Transition Probabilities is Rational

TL;DR

This paper proves that the cover time of a finite-state Markov chain with rational transition probabilities is rational if bounded, and extends this to multiple independent chains, linking cover time to hitting times in higher dimensions.

Contribution

It establishes the rationality of bounded cover times for Markov chains with rational transitions and extends the result to multiple independent chains.

Findings

Bounded cover time is a rational number for chains with rational transition probabilities.

The result applies to multiple independent chains running simultaneously.

The proof relates cover time to hitting times in an extended Markov chain.

Abstract

The cover time of a Markov chain on a finite state space is the expected time until all states are visited. We show that if the cover time of a discrete-time Markov chain with rational transitions probabilities is bounded, then it is a rational number. The result is proved by relating the cover time of the original chain to the hitting time of a set in another higher dimensional chain. We also extend this result to the setting where $k\geq 1 $ independent copies of a Markov chain are run simultaneously on the same state space and the cover time is the expected time until each state has been visited by at least one copy of the chain.