Large deviations of a branching process in random environments with immigration

The report focuses on the study of large deviation probabilities for branching processes in a random environment (BPRE) with immigration, under the assumption that the conditional distribution of the number of direct descendants is geometric, and the steps of the accompanying random walk and random number of immigrants satisfy right-hand Cramer's condition. Two models are considered: in the first, immigration occurs only at the moments of extinction, and in the second - at each moment of time. Random numbers of immigrants are assumed to be a sequence of independent identically distributed random variables. It is also assumed that an expectation of the conditional mean of the direct descendants of a particle is finite. It is shown that, under such restrictions, the asymptotic in two considered cases differs from the similar result for BPRE and random walk by the multiplicative constant.