Quasistationary distributions for level-dependent quasi-birth-and-death processes

Abstract

In this paper we provide a complete quasistationary analysis for the class of level-dependent, discrete-time quasi-birth-and-death processes (QBDs) for which level zero has been collapsed into an absorbing state. We show that the form of a quasistationary distribution depends upon whether the eigenvalue of a certain matrix is equal to one or less than one. Furthermore, we show how to calculate the convergence norm α of such a QBD and observe that the QBD is α-recurrent in the first case mentioned above and α-transient in the second case. The further classification of an α-recurrent QBD as α-positive or α-null depends on whether the convergence radius α2 of the modified QBD in which level one..