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On the ergodicity of a class of level-dependent quasi-birth-and-death processes

Part of: Ergodic theory Markov processes

Published online by Cambridge University Press:  15 November 2019

James D. Cordeiro ,
Jeffrey P. Kharoufeh  and
Mark E. Oxley
James D. Cordeiro*
Affiliation:
University of Dayton
Jeffrey P. Kharoufeh*
Affiliation:
Clemson University
Mark E. Oxley*
Affiliation:
Air Force Institute of Technology
*
*Postal address: Department of Mathematics, University of Dayton, Dayton, OH, USA.
**Postal address: Department of Industrial Engineering, Clemson University, Clemson, SC, USA.
***Postal address: Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, OH, USA.
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Abstract

We examine necessary and sufficient conditions for recurrence and positive recurrence of a class of irreducible, level-dependent quasi-birth-and-death (LDQBD) processes with a block tridiagonal structure that exhibits asymptotic convergence in the rows as the level tends to infinity. These conditions are obtained by exploiting a multi-dimensional Lyapunov drift approach, along with the theory of generalized Markov group inverses. Additionally, we highlight analogies to well-known average drift results for level-independent quasi-birth-and-death (QBD) processes.

Keywords

LDQBD processpositive recurrenceLyapunov potential

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 4 , December 2019 , pp. 1109 - 1128
Copyright
© Applied Probability Trust 2019 

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