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.
Highlights
1.1 Overview Simultaneous renewal is an important topic for a practical application of Markov chains
It has its own value, for example, in queuing theory, we are interested in its investigation because it plays an essential role in coupling construction, which can be used to derive stability estimates of the n-step transition probabilities and other results like the law of large numbers and limit theorems
It worth mentioning that the coupling construction for time-inhomogeneous chains is slightly different from its classical setup
Summary
1.1 Overview Simultaneous renewal is an important topic for a practical application of Markov chains. It has its own value, for example, in queuing theory, we are interested in its investigation because it plays an essential role in coupling construction, which can be used to derive stability estimates of the n-step transition probabilities and other results like the law of large numbers and limit theorems
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