Branching processes with immigration in atypical random environment

\Motivated by a seminal paper of Kesten et al. (Ann. Probab., 3(1), 1-31, 1975) we consider a branching process with a conditional geometric offspring distribution with i.i.d. random environmental parameters A(n), n >= 1 and with one immigrant in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution F of xi(n) := log((1 - A(n))/A(n)) 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 nth generation which becomes even heavier with increase of n. More precisely, we prove that, for all n, the distribution tail P(Z(n) >= m) of the nth population size Z(n) is asymptotically equivalent to n (F) over bar (log m) as m grows. In this way we generalise Bhattacharya and Palmowski (Stat. Probab. Lett., 154, 108550, 2019) who proved this result in the case n = 1 for regularly varying environment F with parameter alpha > 1. Further, for a subcritical branching process with subexponentially distributed xi(n), we provide the asymptotics for the distribution tail P(Z(n) > 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 the presence of a single very small environmental parameter A(k).