A critical Galton–Watson branching process with emigration

In the present paper, an example of critical $\varphi$-branching processes introduced in [1] is investigated. For $\varphi(n)=\max\{0,n-1\}$, we derive an asymptotic formula for the probability $\mathbf P\{\mu(t)>0\mid\mu(0)=m\ge 2\}$ as $t\to\infty$. Here $\mu(t)$ is the number of particles at time $t$. We also obtain a conditional limit theorem for this process which is analogous to a well-known result for a critical Galton–Watson process.