An estimate for an expectation of the simultaneous renewal for time-inhomogeneous Markov chains

TL;DR

This paper derives an upper bound for the expected time until two arbitrary initial distributions of time-inhomogeneous Markov chains simultaneously hit a renewal set, extending previous results to more general conditions.

Contribution

It provides a new upper bound for the expectation of simultaneous renewal times in time-inhomogeneous Markov chains with arbitrary initial distributions.

Findings

Derived an upper bound for the expectation of simultaneous renewal time.

Extended previous results to more general state spaces and initial distributions.

Applied the bound to time-inhomogeneous birth--death Markov chains.

Abstract

In this paper, we consider two time-inhomogeneous Markov chains $X^{(l)}_t$, $l\in\{1,2\}$, with discrete time on a general state space. We assume the existence of some renewal set $C$ and investigate the time of simultaneous renewal, that is, the first positive time when the chains hit the set $C$ simultaneously. The initial distributions for both chains may be arbitrary. Under the condition of stochastic domination and nonlattice condition for both renewal processes, we derive an upper bound for the expectation of the simultaneous renewal time. Such a bound was calculated for two time-inhomogeneous birth--death Markov chains.