A LIMIT THEOREM FOR THE GALTON‐WATSON PROCESS WITH IMMIGRATION

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Abstract
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.

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