随机椭圆云的直径
The diameter of a random elliptical cloud
摘要：
我们研究了独立同分布云的直径或最大点间距离的渐近行为。$d$ -当云中的点数趋于无穷大时的维数随机向量。这是一个非标准极值问题，因为直径是一个最大$U$统计量，因此是一个最大的相关随机变量。因此，极限分布可能不是极值分布。我们得到详尽的结果云的椭圆向量的欧氏直径的欧氏范数是在吸引区域的最大Gumbel分布。我们也得到了结果，在其他规范的球面向量，我们给出了几个二维推广。我们的结果和证明背后的主要思想是随机向量的一个特殊性质，其规范是在吸引区域的Gumbel分布：本地化到子空间的低维向量与一个大的规范。
英文摘要：
We study the asymptotic behavior of the diameter or maximum interpoint distance of a cloud of i.i.d. $d$-dimensional random vectors when the number of points in the cloud tends to infinity. This is a non standard extreme value problem since the diameter is a max-$U$-statistic, hence a maximum of dependent random variables. Therefore, the limiting distributions may not be extreme value distributions. We obtain exhaustive results for the Euclidean diameter of a cloud of elliptical vectors whose Euclidean norm is in the domain of attraction for the maximum of the Gumbel distribution. We also obtain results in other norms for spherical vectors and we give several bi-dimensional generalizations. The main idea behind our results and their proofs is a specific property of random vectors whose norm is in the domain of attraction of the Gumbel distribution: the localization into subspaces of low dimension of vectors with a large norm.
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