Blackwell dominance in large samples

We study repeated independent Blackwell experiments; standard examples include drawing multiple samples from a population, or performing a measurement in different locations. In the baseline setting of a binary state of nature, we compare experiments in terms of their informativeness in large samples. Addressing a question due to Blackwell (1951) we show that generically, an experiment is more informative than another in large samples if and only if it has higher Renyi divergences. As an application of our techniques we in addition provide a novel characterization of k-th order stochastic dominance as second-order stochastic dominance of large i.i.d. sums.