Subcritical branching processes in a random environment without the Cramer condition (Q432502)

The paper investigates a subcritical branching process in a random environment (SBPRE) under the condition that the environment is given by a sequence of i.i.d. (random) probability laws on nonnegative integers. One of the standing assumptions is that \(\text{P}\{X>x\}\) is regularly varying at \(\infty\) with index \(-\beta\), for some \(\beta>2\), where \(X\) denotes the generic step of the associated random walk. Among others, this implies that the SBPRE in focus is neither weakly, nor intermediately, nor strongly subcritical. While the first main result of the paper treats the asymptotics of the survival probability of the process, the second is a Yaglom-type functional conditional limit theorem for the number of particles up to moment \(n\) given the survival to this moment. The heuristics behind both results is that ``the process survives for a long time owing to a single big jump of the associated random walk at the beginning of the evolution which, in turn, is accompanied by an explosion of the population size at this moment''.