Journal of Applied Probability

Quasistochastic matrices and Markov renewal theory

Gerold Alsmeyer


Let S be a finite or countable set. Given a matrix F = (Fij)i,jS of distribution functions on R and a quasistochastic matrix Q = (qij)i,jS, 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≥0QnF*n associated with QF := (qijFij)i,jS (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 QF becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate QF 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.

Article information

J. Appl. Probab., Volume 51A (2014), 359-376.

First available in Project Euclid: 2 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K05: Renewal theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60K15: Markov renewal processes, semi-Markov processes

Quasistochastic matrix Markov random walk Markov renewal equation Markov renewal theorem spread out Stone-type decomposition age-dependent multitype branching process random difference equation perpetuity


Alsmeyer, Gerold. Quasistochastic matrices and Markov renewal theory. J. Appl. Probab. 51A (2014), 359--376. doi:10.1239/jap/1417528486.

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