Multivariate semi-markov matrices

Abstract

Finite matrices with entries p_ij F_ij (x₁,…, x_k), where {p_ij} is stochastic and F_ij(.) is a k-variate probability distribution are discussed. It is shown that the matrix of k-variate Laplace-Stieltjes transforms of the P_ij F_ij(x₁, …, x_k) has a Perron-Frobenius eigenvalue which is a convex function in k variables in a suitably defined region. The values of the partial derivatives near the origin of this maximal eigenvalue are exhibited. They are quantities of interest in a variety of applications in Probability theory.