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This paper is about the asymptotic distribution of linear combinations of stratum means in stratified sampling, with and without replacement. Both the number of strata and their size is arbitrary. Lindeberg conditions are shown to guarantee asymptotic normality and consistency of variance estimators. The same conditions also guarantee the validity of the bootstrap approximation for the distribution of the t-statistic. Via a bound on the Mallows distance, situations will be identified in which the bootstrap approximation works even though the normal approximation fails. Without proper scaling, the naive bootstrap fails.
Bickel et al. (Fri,) studied this question.