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Asymptotic behaviour of critical decomposable 2-type Galton–Watson processes with immigration

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  • Barczy, Mátyás
  • Bezdány, Dániel
  • Pap, Gyula

Abstract

In this paper the asymptotic behaviour of a critical 2-type Galton–Watson process with immigration is described when its offspring mean matrix is reducible, in other words, when the process is decomposable. It is proved that, under second or fourth order moment assumptions on the offspring and immigration distributions, a sequence of appropriately scaled random step processes formed from a critical decomposable 2-type Galton–Watson process with immigration converges weakly. The limit process can be described using one or two independent squared Bessel processes and possibly the unique stationary distribution of an appropriate single-type subcritical Galton–Watson process with immigration. Our results complete and extend the results of Foster and Ney (1978) for some strongly critical decomposable 2-type Galton–Watson processes with immigration.

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  • Barczy, Mátyás & Bezdány, Dániel & Pap, Gyula, 2023. "Asymptotic behaviour of critical decomposable 2-type Galton–Watson processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 318-350.
  • Handle: RePEc:eee:spapps:v:160:y:2023:i:c:p:318-350
    DOI: 10.1016/j.spa.2023.03.003
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    References listed on IDEAS

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    1. Kevei, Péter & Wiandt, Péter, 2021. "Moments of the stationary distribution of subcritical multitype Galton–Watson processes with immigration," Statistics & Probability Letters, Elsevier, vol. 173(C).
    2. Barczy, M. & Ispány, M. & Pap, G., 2011. "Asymptotic behavior of unstable INAR(p) processes," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 583-608, March.
    3. Wei, C. Z. & Winnicki, J., 1989. "Some asymptotic results for the branching process with immigration," Stochastic Processes and their Applications, Elsevier, vol. 31(2), pages 261-282, April.
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