4.2 Article

A Galton-Watson process with a threshold at 1 and an immigration at 0

Journal

STATISTICS & PROBABILITY LETTERS
Volume 201, Issue -, Pages -

Publisher

ELSEVIER
DOI: 10.1016/j.spl.2023.109881

Keywords

Population-size-dependent branching; process; State-dependent immigration; Asymptotic behavior; Moments; Invariant measure; Limit distribution

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This paper investigates a special class of population-size-dependent branching processes, which can be viewed as an extension of Galton-Watson processes with state-dependent immigration. The model considers the size of individuals in each generation and evolves according to different Galton-Watson processes or fixed immigration distributions. By using technical results from Foster (1971) and Pakes (1971), asymptotic results similar to those of Galton-Watson processes with state-dependent immigration are obtained through detailed computations.
In this paper we consider a special class of population-size-dependent branching processes, which can also be seen as an extension of Galton-Watson processes with state-dependent immigration (GWPSDI). The model is formulated as follows. Let Zn be the size of individuals belonging to the nth generation of a population. If Zn > 1, the population evolves as a critical Galton-Watson process with finite variance; if Zn = 1, the population evolves as another Galton-Watson process; if Zn = 0, Zn+1 is drawn from a fixed immigration distribution. Based on the technical routes in Foster (1971) and Pakes (1971), some asymptotic results identical with those of GWPSDI are obtained by detailed computations. & COPY; 2023 Elsevier B.V. All rights reserved.

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