Kemeny's constant for countable Markov chains

In this paper, we aim to investigate Kemeny's constant for discrete-time or continuous-time, irreducible and positive recurrent Markov chains on a countable state space. We show that Kemeny's constant is infinite for a discrete-time Markov chain on an infinitely countable state space, which confirms a conjecture in page 1031 of [3]. Moreover, it is shown that the study on Kemeny's constant for a continuous-time Markov chain with a bounded generator can be converted to that for a discrete-time Markov chain through the uniformization technique. It is different and challenging to analyze Kemeny's constant for a chain with unbounded generator. Finally, a upward skip-free process is considered and some discussions for Kemeny's constant are given. (C) 2020 Elsevier Inc. All rights reserved.