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 a of such a QBD and observe that the QBD is a-recurrent in the first case mentioned above and a-transient in the second case. The further classification of an a-recurrent QBD as a-positive or a-null depends on whether the convergence radius a2 of the modified QBD in which level one is collapsed into an absorbing state is strictly greater than, or equal to, a. In the first of these cases the QBD is a-positive, while in the second case the QBD may be a-positive or a-null

Keywords:

• Quasi-birth-and-death process
• Quasistationary distributions
• Limiting conditional distributions
• β-invariant measures