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Deterministic approximation of a sequence of nearly critical branching processes. (English) Zbl 1154.60070

The author is concerned with a nearly critical sequence of discrete time branching processes with time-dependent immigration. Here, ‘nearly critical’ means that \(a(n) \rightarrow 1\) as \(n\rightarrow \infty\), where \(a(n)\) is the mean number of offspring of a single individual in the \(n\)th process. It is assumed that the sequences of immigration means and variances can be approximated by regularly varying sequences with non-negative exponents. Under suitable assumptions, it is proved that the sequence of branching processes considered can be normalized in such a manner that it converges in the Skorokhod topology to a deterministic process. Consequences for the sequences of maxima and total progenies are also noted.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
62F12 Asymptotic properties of parametric estimators
60G99 Stochastic processes
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References:

[1] Billingsley P., Convergence of Probability Measures (1968) · Zbl 0172.21201
[2] Bingham N.H., Encyclopedia of Mathematics and its Applications 27 (1987)
[3] DOI: 10.1016/0304-4149(94)00076-6 · Zbl 0817.62070
[4] DOI: 10.1239/aap/1118858637 · Zbl 1069.62065
[5] Jacod J., Limit Theorems for Stochastic Processes (2003) · Zbl 1018.60002
[6] DOI: 10.1214/aos/1176325509 · Zbl 0807.62063
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