Multivariate semi-markov matrices

Finite matrices with entries p ij F ij ( x 1 ,…, 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 (x 1 , …, 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.