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

Abstract

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 Q m 0 = P { μ ( t ) > 0 | μ (0) = m } ∼ mQ ( t ) → 0 for any m = 2,3, . . . Let for any integer m > r ≥ 1 In this paper, we prove that Q mr ( t ) ∼ ( m − r ) Q ( t ) as t → ∞ for any critical continuous-time Markov branching process. Earlier, this result was obtained for branching processes with finite variation of the number of particles.