Functional limit theorems for Galton–Watson processes with very active immigration

Abstract

We prove weak convergence on the Skorokhod space of Galton-Watson processes with immigration, properly normalized, under the assumption that the tail of the immigration distribution has a logarithmic decay. The limits are extremal shot noise processes. By considering marginal distributions, we recover the results of Pakes [Adv. Appl. Probab., 11(1979), 31-62].

Limit theorems for continuous-state branching processes with immigration

“…We refer also to the references therein for previous works on this topic, see in particular Pakes [Pak79] and Pinsky [Pin72]. Recently, a functional limit theorem for GWIs has been established by Iksanov and Kabluchko, see [IK18]. They found an interesting regime of immigration for which the limiting process arising is an extremal shot noise process.…”

“…This will enable us to deal with the possibility that the process Y hits 0. The convergence of point processes explained in the heuristics is the route chosen in [IK18] for dealing with the log-case for GWIs. We shall take another path and work with generators.…”

On central limit theorems for branching processes with dependent immigration

“…More precisely, they studied weak convergence of n −1 W (n) under more general situation that there exists a random vari- Badalbaev and Zubkov [5] for the sequence of special random processes (including branching processes with immigration) proved a limit theorem which contains results of [29] and [1] as a special case. Concerning functional limit theorems for (1), we refer to Wei and Winnicki [46], Sriram [43], Li [25], [24], Ispány et al [14], [15], Khusanbaev [19], [21], Iksanov and Kabluchko [13] and see references therein. We refer to papers [18], [20], [39] where the rates of convergence in central limit theorem for (1) were studied.…”