QUASISTOCHASTIC MATRICES AND MARKOV RENEWAL THEORY

Let S be a finite or countable set. Given a matrix F = (F-ij)(i), (j is an element of S) of distribution functions on R and a quasistochastic matrix Q = (qij)(i), (j is an element of 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 Sigma(n >= 0) Q(n) circle times F*(n) associated with Q circle times F := (q(ij)F(ij))(i, j) (is an element of 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, andWiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q circle times F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q circle times F to a Markov random walk {(M-n, S-n)}(n >= 0) with discrete recurrent driving chain {M-n}(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.