Journal of Applied Mathematics and Stochastic Analysis
Volume 4 (1991), Issue 4, Pages 263-292
Conditional limit theorems for branching processes
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.
Let be a branching process in which each
individual reproduces independently of the others and has probability of giving rise to descendants in the following generation.
The random variable is the number of individuals in the th
generation. It is assumed that . Denote by the total
progeny, , the time of extinction, and , the total number of ancestors
of all the individuals in the process. This paper deals with the
distributions of the random variables , and under the condition
that and determines the asymptotic behavior of these distributions
in the case where and in such a way that tends to a
finite positive limit.