Functional limit theorems for Galton-Watson processes with very active   immigration
  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]. 
Alexander Iksanov and Zakhar Kabluchko
math.PR