Fluctuation Limit of Branching Processes with Immigration and Estimation of the Means

A sequence of Galton–Watson branching processes with immigration is investigated, when the offspring mean tends to its critical value one and the offspring variance tends to zero. It is shown that the fluctuation limit is an Ornstein– Uhlenbeck type process. As a consequence, in contrast to the case where the offspring variance tends to a positive limit, the conditional least squares estimator of the offspring mean turns out to be asymptotically normal. The norming factor is n, in contrast to the subcritical case where it is n, and in contrast to the nearly critical case with positive limiting offspring variance, where it is n.