Let [ξ(m),m=0,1,2,…] be a branching process in which each
individual reproduces independently of the others and has probability pj(j=0,1,2,…) of giving rise to j descendants in the following generation.
The random variable ξ(m) is the number of individuals in the mth
generation. It is assumed that P{ξ(0)=1}=1. Denote by ρ the total
progeny, μ, the time of extinction, and τ, the total number of ancestors
of all the individuals in the process. This paper deals with the
distributions of the random variables ξ(m), μ and τ under the condition
that ρ=n and determines the asymptotic behavior of these distributions
in the case where n→∞ and m→∞ in such a way that m/n tends to a
finite positive limit.