Conditional limit theorems for branching processes

Let [ ξ ( m ) , m = 0 , 1 , 2 , … ] be a branching process in which each individual reproduces independently of the others and has probability p j ( 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 m th 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.