An investigation of the maximal number of offspring in galton-watson processes with immigration

الملخص EN

Consider a Galton-Watson process in which each individual reproduces independently of all others with probability Pk, k = 0, 1, 2, ...

of giving rise to j progenies in the following generation and in which there is an independent immigration component.

We have investigated the asymptotic behavior of the process with immigration in critical, subcritical and supercritical case.

In critical and supercritical cases the random variable Wn which is the maximal number of direct offspring in a generation, converges with probability one to ∞, as n→∞ , whereas when normalizing the process in critical case, with a <∞and f11(l) < ∞, Wn has a limiting distribution as n.

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