GENERAL SOLUTION OF THE POISSON EQUATION FOR QUASI-BIRTH-AND-DEATH PROCESSES.

We consider the Poisson equation (I -- P)u = g, where P is the transition matrix of a quasi-birth-and-death process with infinitely many levels, g is a given infinite dimensional vector, and u is the unknown. Our main result is to provide the general solution of this equation. To this purpose we use the block tridiagonal and block Toeplitz structure of the matrix P to obtain a set of matrix difference equations, which are solved by constructing suitable resolvent triples. [ABSTRACT FROM AUTHOR]