We study supercritical branching processes under the influence of an independent and identically distributed (i.i.d.) emigration component. We provide conditions under which the lifetime of the process is finite or has a finite expectation. A theorem of Kesten–Stigum type is obtained, and the extinction probability for a large initial population size is related to the tail behaviour of the emigration.

In Lyons, Pemantle and Peres (1995), a martingale change of measure method was developed in order to give an alternative proof of the Kesten–Stigum L log L theorem for single-type branching processes. Later, this method was extended to prove the L log L theorem for multiple- and general multiple-type branching processes in Biggins and Kyprianou (2004), Kurtz et al. (1997), and Lyons (1997). In this paper we extend this method to a class of superdiffusions and establish a Kesten–Stigum L log L type theorem for superdiffusions. One of our main tools is a spine decomposition of superdiffusions, which is a modification of the one in Englander and Kyprianou (2004).

Through the study of a simple embedded martingale we obtain an extension of the Kesten–Stigum theorem and prove a central limit theorem for controlled Galton-Watson processes.

An analogue of the Kesten–Stigum theorem, and sufficient conditions for the geometric rate of growth in the rth mean and almost surely, are obtained for population-size-dependent branching processes.