The asymptotic behaviour of the non-extinction probability of a bounded from below continuous time Markov critical branching process with ininite variance

摘要

Let μ(t) be the number of particles at time t of a continuous-time critical branching process. It is known that the probability of non-extinction of the process at time t Q(t)=P{μ(t)>0|μ(0)=1}→0 as t →∞, hence it follows that Qmo=P{μ(t)>0|μ(0)=m}∽mQ(t)→0 for any m =2,3,…Let for any integer m>r≥1 Q_mr(t)=P{inf_(0≤u≤t) μ(u)>r|μ(0)=m}.In this paper, we prove that Qmr(t)~(m-r)Q(t)as t →∞ co for any critical continuous-time Markov branching process. Earlier, this result was obtained for branching processes with finite variation of the number of particles.