  In this paper, a critical Galton-Watson branching process with immigration
$Z_{n}$ is studied. We first obtain the convergence rate of the harmonic moment
of $Z_{n}$. Then the large deviation of $S_{Z_n}:=\sum_{i=1}^{Z_n} X_i$ is
obtained, where $\{X_i\}$ is a sequence of independent and identically
distributed zero-mean random variables with tail index $\alpha>2$. We shall see
that the converging rate is determined by the immigration mean, the variance of
reproducing and the tail index of $X_1^+$, comparing to previous result for
supercritical case, where the rate depends on the Schr\"{o}der constant and the
tail index.
