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

摘要

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 theexistence of some renewal set $C$ and investigate the time of simultaneousrenewal, 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 bothrenewal processes, we derive an upper bound for the expectation of thesimultaneous renewal time. Such a bound was calculated for twotime-inhomogeneous birth--death Markov chains.