## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 19, Number 4 (1991), 1640-1663.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176990227

Digital Object Identifier
doi:10.1214/aop/1176990227

Mathematical Reviews number (MathSciNet)
MR1127719

Zentralblatt MATH identifier
0746.60030

JSTOR
Kinoshita, K.; Resnick, Sidney I. Convergence of Scaled Random Samples in $\mathbb{R}^d$. Ann. Probab. 19 (1991), no. 4, 1640--1663. doi:10.1214/aop/1176990227. https://projecteuclid.org/euclid.aop/1176990227