Fluctuation limit of branching processes with immigration and estimation of the means

We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n3/2 , in contrast to both the subcritical case, in which it is n1/2, and the nearly critical case with positive limiting offspring variance, in which it is n.