Large Deviation Rates for Supercritical and Critical Branching Processes

This paper investigates large deviation results for the supercritical multi-type and the critical single type branching processes when conditioned on non extinction thus extending the results in [1] and [2] in two directions. We show for example that in the multitype supercritical case the probability of large deviation between the empirical population proportion and its stable limit decays geometrically. Similarly in the critical single type case the (large deviation) probability that the ratio of the population at time (n + 1) to that at time n deviates from one by more than decays at an algebraic rate. A number of similar results are presented here. Some open problems are indicated.