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Quasistochastic matrices and Markov renewal theory

Part of: Special processes Markov processes

Published online by Cambridge University Press:  30 March 2016

Gerold Alsmeyer
Gerold Alsmeyer*
Affiliation:
Institute of Mathematical Statistics, Department of Mathematics and Computer Science, University of Münster, Orléans-Ring 10, D-48149 Münster, Germany. Email address: gerolda@math.uni-muenster.de.
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Abstract

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Let 𝓈 be a finite or countable set. Given a matrix F = (Fij)i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (qij)i,j∈𝓈, i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑n≥0Qn ⊗ F*n associated with Q ⊗ F := (qijFij)i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(Mn, Sn)}n≥0 with discrete recurrent driving chain {Mn}n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

Keywords

Quasistochastic matrixMarkov random walkMarkov renewal equationMarkov renewal theoremspread outStone-type decompositionage-dependent multitype branching processrandom difference equationperpetuity

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J45: Probabilistic potential theory 60K15: Markov renewal processes, semi-Markov processes
Type
Part 8. Markov processes and renewal theory
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 359 - 376
Copyright
Copyright © Applied Probability Trust 2014 

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