On the convergence of scaled random samples (Q2565365)

The scaled-sample problem is considered: Let \(\{X^{(j)}, j\geq 1\}\) be an i.i.d. sequence of random elements with values in a normed linear space \(E\). When do there exist scaling constants \(\{\gamma_n\}\) such that \(\{X^{(j)}/\gamma_n, 1\leq j\leq n\}\) converges as \(n\to\infty\) (in the Hausdorff metric given by the norm) to a fixed set \(K\)? The main result relates the convergence of scaled samples to a large deviation principle for single observations. Some examples are also given.