An Asymptotic Representation Theorem

Abstract

Let a sequence of statistical experiments converge to a limit experiment in the sense of Le Cam (1972). Furthermore assume that a sequence of statistics possesses a limit distribution under every statistical parameter. Then its set of limit distributions is also the set of distributions of some randomized estimator in the limit experiment. This is a simple way of saying that the limit experiment is a 'lower bound' for the converging sequence of experiments. Moreover, convolution and minimax theorems can be obtained as corollaries.