(untitled) (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 2015)
In this communication it is proved a fluctuation limit theorem for controlled branching processes. Under the conditions that the offspring and control means tend to be critical, the obtained limit is a diffusion process. This result is applied to conclude that the standard parametric bootstrap weighted conditional least squares estimate for the offspring mean is asymptotically invalid in the critical case.