  In this paper, we consider a critical Galton-Watson branching process with
immigration stopped at zero $\mathbf{W}$. Some precise estimation on the
generation function of the $n$-th population are obtained, and the tail
probability of the life period of $\mathbf{W}$ is studied. Based on above
results, two conditional limit theorems for $\mathbf{W}$ are established.
