Heavy-Tailed Branching Process with Immigration

Abstract

In this article, we analyze a branching process with immigration defined recursively by random variables independent of B t . We assume that one of generic variables A and B has a regularly varying tail distribution. We identify the tail behavior of the distribution of the stationary solution X t .We also prove CLT for the partial sums that could be further generalized to FCLT. Finally, we also show that partial maxima have a Fréchet limiting distribution.

Regularly varying non-stationary Galton–Watson processes with immigration

“…Galton-Watson processes with immigration have been frequently used for modeling the sizes of a population over time, so a delicate description of their tail behavior is an important question. In this paper we focus on regularly varying not necessarily stationary Galton-Watson processes with immigration, complementing the results of Basrak et al [2] for the stationary case. By a Galton-Watson process with immigration, we mean a stochastic process (X n ) n 0 given by…”

On Aggregation of Subcritical Galton–Watson Branching Processes with Regularly Varying Immigration

“…Indeed, as shown by Basrak et al[6, Lemma 3.1], (Y ) ∈Z+ is the forward tail process of (X ) ∈Z . On the other hand, by Janssen and Segers[12, Ex.…”

“…In Appendix C, we formulate a result on weak convergence of partial-sum processes toward Lévy processes by slightly modifying Theorem 7.1 in Resnick [25] with a different centering. In the end, we recall a result on the tail behavior and forward tail process of (X k ) k 0 due to Basrak et al [6], and we determine the limit measures of finite segments of (X k ) k 0 ; see Appendix D.…”

“…Let (X k ) k∈Z be a strongly stationary extension of (X k ) k∈Z+ . Basrak et al [6,Lemma 3.1] described the socalled forward tail process of the strongly stationary process (X k ) k∈Z , and hence, due to Basrak and Segers [7,Thm. 2.1], the strongly stationary process (X k ) k∈Z is jointly regularly varying.…”

On aggregation of multitype Galton–Watson branching processes with immigration

“…In Appendix D we formulate a result on weak convergence of partial sum processes towards Lévy processes by slightly modifying Theorem 7.1 in Resnick [25] with a different centering. In the end, we recall a result on the tail behavior and forward tail process of (X k ) k 0 due to Basrak et al [5], and we determine the limit measures of finite segments of (X k ) k 0 , see Appendix E.…”