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

Dawei Lu and Huangding Lv

Statistics & Probability Letters, 2023, vol. 201, issue C

Abstract: 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.

Keywords: Population-size-dependent branching process; State-dependent immigration; Asymptotic behavior; Moments; Invariant measure; Limit distribution (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spl.2023.109881

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