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.
Publié le : 1991-10-14
Classification:
random sets,
extremes,
regular variation,
upper semicontinuous functions,
almost sure convergence,
60F15,
60B05
@article{1176990227,
author = {Kinoshita, K. and Resnick, Sidney I.},
title = {Convergence of Scaled Random Samples in $\mathbb{R}^d$},
journal = {Ann. Probab.},
volume = {19},
number = {4},
year = {1991},
pages = { 1640-1663},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990227}
}
Kinoshita, K.; Resnick, Sidney I. Convergence of Scaled Random Samples in $\mathbb{R}^d$. Ann. Probab., Tome 19 (1991) no. 4, pp. 1640-1663. http://gdmltest.u-ga.fr/item/1176990227/