2020/07/27 by Foss, Sergey, Korshunov, Dmitry, Palmowski, Zbigniew
#60G55 #60J70 #60J80 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2007.13507
Motivated by a seminal paper of Kesten et al. (1975) we consider a branching process with a geometric offspring distribution with i.i.d. random environmental parameters An, n≥ 1 and size -1 immigration in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution F of ξn := log ((1-An)/An) is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n-th generation which becomes even heavier with increase of n. More precisely, we prove that, for any n, the distribution tail ℙ(Zn > m) of the n-th population size Zn is asymptotically equivalent to nF(log m) as m grows. In this way we generalize Bhattacharya and Palmowski (2019) who proved this result in the case n=1 for regularly varying environment F with parameter α>1. Further, for a subcritical branching process with subexponentially distributed ξn, we provide the asymptotics for the distribution tail ℙ(Zn>m) which are valid uniformly for all n, and also for the stationary tail distribution. Then we establish the "principle of a single atypical environment" which says that the main cause for the number of particles to be large is a presence of a single very small environmental parameter Ak.