A conditional limit theorem for a critical Branching process with immigration

The life period of a branching process with immigration begins at the moment $T$ and has length $\tau$ if the number of particles $\mu(T-0)=0$, $\mu(t)>0$ for all $T\le t<T+\tau$, $\mu(T+\tau)=0$ (the trajectories of the process are assumed to be continuous from the right). For a critical Markov branching process is obtained a limit theorem on the behavior of $\mu(t)$ under the condition that $\tau>t$ and $T=0$.