A Fluctuation Limit Theorem of Branching Processes with Immigration and Statistical Applications

We prove a general fluctuation limit theorem for Galton-Watson branching processes with immigration. The limit is a time-inhomogeneous OU type process driven by a spectrally positive Levy process. As applications of this result, we obtain some asymptotic estimates for the conditional least-squares estimator of the offspring means and variances of the offspring and immigration distributions.