Convergence of Scaled Random Samples in $\mathbb{R}^d$

Abstract

Let $\{\mathbf{X}_j, 1 \leq j \leq n\}$ be a sequence of iid random vectors in $\mathbb{R}^d$ and $S_n = \{\mathbf{X}_j/b_n, 1 \leq j \leq n\}$. When do there exist scaling constants $b_n \rightarrow \infty$ such that $S_n$ converges to some compact set $S$ in $\mathbb{R}^d$ almost surely (in probability)? We show that a limit set $S$ is star-shaped (i.e., $\mathbf{x} \in S$ implies $t\mathbf{x} \in S$, for $0 \leq t \leq 1$) so that after a polar coordinate transformation the limit set is the hypograph of an upper semicontinuous function. We specify necessary and sufficient conditions for convergence to a particular limit set. Some examples are also given.