Asymptotic behaviour of the S-stopped branching processes with countable   state space
  he starting process with countable number of types \mu(t) generates a stopped branching process \xi(t). The starting process stops, by falling into the nonempty set S. It is assumed, that the starting process is subcritical, indecomposable and noncyclic. It is proved, that the extinction probability converges to the cyclic function with period 1. 
Iryna Kyrychynska, Ostap Okhrin, Yaroslav Yeleyko
math.ST stat.TH