Estimation of the offspring mean of a supercritical or near-critical size-dependent branching process

We consider a single-type supercritical or near-critical size-dependent branching process Nn n such that the offspring mean converges to a limit m1 with a rate of convergence of order as the population size Nn grows to 8 and the variance may change at the rate , where a>0 and -1ß0 et -1