Conditional limit theorems for branching processes

Lajos Takács

International Journal of Stochastic Analysis, 1991, vol. 4, 1-30

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.

Date: 1991
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:614917

DOI: 10.1155/S1048953391000217

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