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The main subject of the study in this paper is the simultaneous renewal timefor two time-inhomogeneous Markov chains which start with arbitrary initialdistributions. By a simultaneous renewal we mean the first time of jointhitting the specific set $C$ by both processes. Under the condition ofexistence a dominating sequence for both renewal sequences generated by thechains and non-lattice condition for renewal probabilities an upper bound forthe expectation of the simultaneous renewal time is obtained.

On estimation of expectation of simultaneous renewal time of time-inhomogeneous Markov chains using dominating sequence