Journal of Applied Mathematics and Stochastic Analysis
Volume 4 (1991), Issue 4, Pages 263-292
doi:10.1155/S1048953391000217

Conditional limit theorems for branching processes

Lajos Takács1,2

1Case Western Reserve University, Cleveland, Ohio, USA
22410 Newbury Drive, Cleveland Heights 44118, OH, USA

Received 1 August 1991; Revised 1 September 1991

Copyright © 1991 Lajos Takács. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.