## Journal of Applied Probability

### Quasistochastic matrices and Markov renewal theory

Gerold Alsmeyer

#### Abstract

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

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

Dates
First available in Project Euclid: 2 December 2014

https://projecteuclid.org/euclid.jap/1417528486

Digital Object Identifier
doi:10.1239/jap/1417528486

Mathematical Reviews number (MathSciNet)
MR3317369

Zentralblatt MATH identifier
1325.60140

#### Citation

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

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