A limit theorem of branching processes and continuous state branching processes

C" -( S ) be the set of a ll strictly positive continuous functions on S.') It is easy to see that, for each f ( S ) , the function ví (A) defined by 1) i.e., bounded Borel measures.2) T h e topology o f So is that o f w e a k convergence: S o i s compact metrizable by this topology.3) Cf.Bourbaki [1] Chap. 9, p. 44.4) This notation, which is slightly different from t h e usual one, is more convenient in future discussions.1 1 , 1 and 1 / P 2 , the composition V P 3 = 4 P 1 ( 1 / P 2 ) is defined byLemma 1. 2. I f qp i lv, i=1, 2 , then *1 (Vr2) E r .P r o o f .For any ,u and lip E r, S s *(x; f),u(dx) E T .Therefore given Ili-, i= 1 , 2 , and ,u, there exists a unique P E f f t such that e x p ( --, * 1 ( x ; f )/..1(d x ))= s e p(da).