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In this paper, we present a random variable to which the Bernoulli type Galton–Watson branching process with immigration converges in law. This convergence is based on the limiting distribution of the process given in [Commun. Stoch. Anal. 5 (2011), no. 3, 457–480]. We also give another proof of the result in [Commun. Stoch. Anal. 5 (2011), no. 3, 457–480] in this paper.

Asymptotic behavior of the Bernoulli type Galton–Watson branching process with immigration