The critical branching process with immigration stopped at zero

Abstract

Limit theorems for the Galton–Watson process with immigration (BPI), where immigration is not permitted when the process is in state 0 (so that this state is absorbing), have been studied for the subcritical and supercritical cases by Seneta and Tavaré (1983). It is pointed out here that, apart from a change of context, the corresponding theorem in the critical case has been obtained by Vatutin (1977). Extensions which follow from a more general form of initial distribution are sketched, including a new form of limit result (7).