Diffusion Approximation of an Array of Controlled Branching Processes

In this paper an array of controlled branching processes is considered. Using operator semigroup convergence theorems, it is proved that the fluctuation limit is a diffusion process under the conditions that the offspring and control means tend to be critical. As an application of this result, in a parametric framework, it is obtained that the bootstrapping distribution of the weighted conditional least squares estimator of the offspring mean in the critical case is not consistent. From this, it is concluded that the standard parametric bootstrap weighted conditional least squares estimate is asymptotically invalid in the critical case.