A note on the state occupancy distribution for Markov chains

Abstract: In a paper, Shah [arXiv:2502.03073] derived an explicit expression for the distribution of occupancy times for a two-state Markov chain, using a method based on enumerating sample paths. We consider here the more general problem of finding the distribution of occupancy times for countable-state Markov chains in discrete time. Our approach, which employs generating functions, leads to arguably simpler formulae for the occupancy distribution for the two-state chain.