Some Limit Theorems for the Critical Galton–Watson Branching Processes

We consider the critical Galton–Watson processes starting from a random number of particles and determine the effect of the mean value of initial state on the asymptotic state of the process. For processes starting from large numbers of particles and satisfying condition ( S ), we prove the limit theorem similar to the result obtained by W. Feller. We also prove the theorem under the condition W ( n ) > 0 for the critical processes satisfying the conditions ( S ) and ( M ).