On the convergence of scaled random samples

Abstract

The scaled-sample problem asks the following question: given a distribution on a normed linear space E, when do there exist constants ${\gamma_n} such that $X^{(j)}/\gamma_n}_{j=1}^n$ converges as $n \to \infty$ (in the Hausdorff metric given by the norm) to a fixed set K? (Here ${X^{(j)}}$ are i.i.d. with the given distribution). The main result presented here relates the convergence of scaled samples to a large deviation principle for single observations, thereby achieving a dimension-free description of the problem.