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A LIMIT THEOREM FOR THE GALTON‐WATSON PROCESS WITH IMMIGRATION
Author(s) -
Quine M. P.,
Seneta E.
Publication year - 1969
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1969.tb00104.x
Subject(s) - mathematics , limit (mathematics) , distribution (mathematics) , asymptotic distribution , central limit theorem , gamma distribution , statistical physics , mathematical analysis , statistics , physics , estimator
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.