A LIMIT THEOREM FOR THE GALTON-WATSON PROCESS WITH IMMIGRATION

Summary It is difficult, in general, to optain an explicit expression for the limiting-stationary distribution, when such a distribution exists, of the process in which teh individuals reproduce as in a Galton-Wastson process, but are also subject to an independent immigration component at each generation. The main result of this paper is a limit theorem which suggests a means of approximating this distribution by a gamma density, when the mean of the offspring distribution is less than, but close to, unity. Following along the same lines, it is easy to show that a similar limit theorem holds for the asymptotic conditional limit distribution of an ordinary subcritical Galton-Watson process, whereby this distribution approaches the exponential as the offspring mean approaches unity.