Small positive values for supercritical branching processes in random environment (Q405490)

For \((Z_n)_{n\in \mathbb{N}_0}\) a supercritical branching process in a random environment, the authors investigate small value probabilities. The main result of the paper states that the limit \(\rho:=\lim_{n\to\infty}n^{-1}\operatorname{P}\{Z_n=j|Z_0=k\}\) exists, is nonnegative, and does not depend on \(k\) and \(j\) provided these belong to a certain set \(A\), say, of positive integers. A formula for \(\rho\) is also given as well as sufficient conditions that ensure that \(\rho\) is positive. Under natural assumptions, a useful upper bound for \(\rho\) is found. The authors note that the main result may fail to hold if \(k\) and \(j\) do not belong to \(A\). To illustrate this point, refined examples are given at the end of the paper.NEWLINENEWLINESuppose from now on that the reproduction laws are linear fractional. Then, an explicit expression for \(\rho\) is pointed out which is considerably simpler than the general formula for \(\rho\). It turns out that on the level of large deviations there are two situations: the rare event \((Z_n=k)\) is explained either by the randomness of the environment alone, or by demographical stochasticity. The influence of this dichotomy is discussed for the asymptotics of the most recent common ancestor for a population conditioned to be small but positive at large times.