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A Markov renewal process imbedded in a Markov chain

Petar M. Todorovic, Joe Gani

Open publisher page 4 citations

Abstract

Let be a Markov chain with state space in R + = (0,∞), an initial distribution μ and a transition probability Q. For each x∊R + the support of is [0,x], which implies that . Set and put . We prove that is a Markov renewal process and that is a Markov process with a stationary transition probability function. Write and suppose that . We give conditions under which is relatively stable and show that , where are stabilizing constants and Z is exponentially distributed. We also show that is asymptotically stationary and possesses a mixing property

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What this paper is about

Let be a Markov chain with state space in R + = (0,∞), an initial distribution μ and a transition probability Q. For each x∊R + the support of is [0,x], which implies that . Set and put . We prove that is a Markov renewal process and that is a Markov process with a stationary transition probability function. Write and suppose that . We give conditions under which is relatively stable and show that , where are stabilizing constants and Z is exponentially distributed. We also show that is asymptotically stationary and possesses a mixing property

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Available abstract

Let be a Markov chain with state space in R + = (0,∞), an initial distribution μ and a transition probability Q. For each x∊R + the support of is [0,x], which implies that . Set and put . We prove that is a Markov renewal process and that is a Markov process with a stationary transition probability function. Write and suppose that . We give conditions under which is relatively stable and show that , where are stabilizing constants and Z is exponentially distributed. We also show that is asymptotically stationary and possesses a mixing property

Key concepts: Markov chain, Mathematics, Markov renewal process, Markov property, Continuous-time Markov chain, Mixing (physics), Markov process, Markov chain mixing time

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