Asymptotics of the survival probability of a bounded from below Markov critical branching process with continuous time and infinite variance

Let $\mu(t)$ 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 $t$
$$ Q(t)=\boldsymbol{\mathsf P}\{\mu(t)>0\mid \mu(0)=1\}\to 0 $$
as $t\to\infty$. Hence it follows that
$$ Q_{m0}=\boldsymbol{\mathsf P}\{\mu(t)>0\mid \mu(0)=m\}\sim m Q(t)\to 0 $$
for any $m=2,3,\dotsc$ Let for any integer $m>r\geq1$
$$ Q_{mr}(t)=\boldsymbol{\mathsf P}\{\inf_{0\leq u\leq t}\mu(u)>r\mid\mu(0)=m\}. $$
In this paper, we prove that
$$ Q_{mr}(t)\sim (m-r)Q(t) $$
as $t\to\infty$ for any critical continuous-time Markov branching process. Earlier, this result was obtained for branching processes with finite variation of the number of particles.
This research was supported by the Russian Foundation for Basic Research, grant 05.01.00035, and by the program of the President of Russian Federation for support of leading scientific schools, grant 1758.2003.1.