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

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