Quasistochastic matrices and Markov renewal theory

Abstract

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.