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What can be said about the convergence rate of stratified sample means to a normal distribution, given different allocation schemes? Obviously, under very poor allocation, this convergence can fail (for instance, keeping the sample size in one stratum fixed); but for any reasonable scheme, and under reasonable hypotheses, a CLT applies (Bickel and Freedman 1984, etc). However, is anything known about the sample sizes required to obtain close approximations to a normal distribution, given, say, a reasonable allocation scheme, and information about the (sample size needed for) normal approximation under simple random sampling from the distribution? Or at least for proportional and/or optimal allocation?

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