Integer-valued branching processes with immigration

The notion of self-decomposability for N0-valued rv's as introduced by Steutel and van Harn [8] and its generalization by van Ham, Steutel and Vervaat [4], are used to study the limiting behaviour of continuous-time branching processes with immigration. This behaviour provides analogues to the behaviour of sequences of rv's obeying a certain difference equation as studied by Vervaat [10] and their continuous-time counterpart considered by Wolfe [11]. Furthermore, discrete-state analogues are given for results on stability in the processes studied by Wolfe, and for results on self-decomposability in supercritical branching processes by Yamazato [12].