Open Access
October, 1991 Convergence of Scaled Random Samples in $\mathbb{R}^d$
K. Kinoshita, Sidney I. Resnick
Ann. Probab. 19(4): 1640-1663 (October, 1991). DOI: 10.1214/aop/1176990227


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.


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K. Kinoshita. Sidney I. Resnick. "Convergence of Scaled Random Samples in $\mathbb{R}^d$." Ann. Probab. 19 (4) 1640 - 1663, October, 1991.


Published: October, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0746.60030
MathSciNet: MR1127719
Digital Object Identifier: 10.1214/aop/1176990227

Primary: 60F15
Secondary: 60B05

Keywords: Almost sure convergence , Extremes , random sets , regular variation , upper semicontinuous functions

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 4 • October, 1991
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