Relation between Hitting Times and Probabilities for Imprecise Markov Chains

TL;DR

This paper explores the relationship between hitting times and probabilities in imprecise Markov chains, establishing key implications and equivalences under certain conditions, with a counterexample illustrating limitations.

Contribution

It introduces a formal framework for analyzing hitting times and probabilities in IMCs with specific transition matrix sets, proving new implications and equivalences.

Findings

Finiteness of upper expected hitting time implies lower hitting probability equals one.

Finiteness of lower expected hitting time implies upper hitting probability equals one.

Counterexample shows the converse of the second implication can fail.

Abstract

In the present paper, we investigate the relationship between hitting times and hitting probabilities in discrete-time imprecise Markov chains (IMCs). We define lower and upper hitting times and probabilities for IMCs whose set of transition matrices $\T$ is compact, convex, and has separately specified rows. Building on reachability-based partitions of the state space, we prove two key implications: (i) finiteness of the upper expected hitting time entails the lower hitting probability equals one, and (ii) finiteness of the lower expected hitting time entails the upper hitting probability equals one. We further show an equivalence: the upper expected hitting time is finite if and only if the lower hitting probability is one. Finally, by presenting a counterexample, we show that the converse of the second implication can fail.