Conditional limit theorems for branching processes. (1991) - Takács, Lajos
Abstract : 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.