On the asymptotic behaviour of subcritical branching processes with continuous state space

Summary A detailed account is given of the asymptotic behaviour, in the subcritical case, of a simple class of continuous-state branching processes. In particular, a recent theorem of Kuczma from the theory of functional equations is used to obtain necessary and sufficient conditions for theorems of Kolmogorov-Yaglom type to hold, thus strengthening, for the class of processes treated, some general results of Jirina. It is also shown that these results are closely related to a theorem on the convergence of sums of independent branching processes, and to certain properties of infinitely divisible distributions on the non-negative reals. The theorems are obtained in the first place for processes with unit initial quantity of ancestor, but in a final section it is shown that they can be extended to processes with a general initial distribution if, and only if, this distribution satisfies a regular variation condition.