Conditional limit theorems for branching processes

Let be a branching process in which each individual reproduces independently of the others and has probability of giving rise to j descendants in the following generation. The random variable is the number of individuals in the mth generation. It is assumed that . Denote by the total progeny, , the time of extinction, and , the total number of ancestors of all the individuals in the process. This paper deals with the distributions of the random variables , and under the condition that and determines the asymptotic behavior of these distributions in the case where and in such a way that tends to a finite positive limit.