Bootstrap of means under stratified sampling

In a two-stage cluster sampling procedure, $n$ random populations are drawn independently from independent populations and a sub-sample of observations is taken in each of them. The estimator of the general mean of the observed variables is asymptotically Gaussian and the asymptotic distributions of several bootstrap versions of the normalized and studentized statistics are studied. A weighted population resampling provides a good approximation and its accuracy depends on the convergence rate of the sample size of the populations.