An estimate for an expectation of the simultaneous renewal for time-inhomogeneous Markov chains
Volume 3, Issue 4 (2016), pp. 315–323
Pub. online: 23 December 2016
Type: Research Article
Open Access
Open Access
Received
28 November 2016
28 November 2016
Revised
6 December 2016
6 December 2016
Accepted
7 December 2016
7 December 2016
Published
23 December 2016
23 December 2016
Abstract
In this paper, we consider two time-inhomogeneous Markov chains ${X_{t}^{(l)}}$, $l\in \{1,2\}$, 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.
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