Laws of large numbers for stochastic orders
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We establish laws of large numbers for comparing sums of i.i.d. random variables in terms of stochastic dominance. Our results shed new light on a classic question, raised first by Samuelson (1963), on the relation between expected utility, risk aversion, and the aggregation of independent risks. In the context of statistical experiments, we answer a long-standing open question posed by 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.
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