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Vitaliy Golomoziy

Taras Schevchenko National University of Kyiv, Kyiv, Ukraine

**Part of the book:Â **Stochastic Processes: Fundamentals and Emerging Applications

**Chapter DOI:Â **https://doi.org/10.52305/RXVO2856

This chapter demonstrates how the coupling method can be adapted to study stability of time-inhomogeneous Markov chains. We consider two Markov chains (ê“_{n}(Â¹) )_{n}â‰¥_{0} and (X_{n}(Â²) )_{n}â‰¥_{o }which are discrete-time, time-inhomogeneous, independent and having values in a general phase space. We show that, assuming some proximity of the transition kernels for the chains X(Â¹) and X(Â²), we can modify the standard coupling technique to establish the proximity of the n-step transition probabilities. We show how to obtain estimates for such proximity and calculate it explicitly under conditions similar to uniform ergodicity conditions for the homogeneous Markov chain.

**Keywords:** inhomogeneous Markov chain, coupling, renewal

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[8] Golomoziy V. and Kartashov M. V., Maximal coupling and stability of discrete nonhomogeneous Markov chains, Theory of Probability and Mathematical Statistics, vol.

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[13] Golomoziy V., On estimation of expectation of simultaneous renewal time of timeinhomogeneous Markov chains using dominating sequence, Modern Stochastics: Theory and Applications, vol. 6, no. 3, pp. 333-343, 2019. DOI:10.15559/19-VMSTA138.

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[15] Golomoziy V., An inequality for the coupling moment in the case of two inhomogeneous markov chains, Theory of Probability and Mathematical Statistics, vol. 90, pp.

43-56, 2015. DOI:10.1090/tpms/948

We publish over 800 titles annually by leading researchers from around the world. Submit a Book Proposal Now!