Multidimensional point processes of extreme order statistics

Extremes in Multivariate Mixing Sequences

“…Extremes and exceedances of high boundaries by stationary and nonstationary multivariate random sequences in the rare events context were presented by Falk et al [2]. The purpose of this paper is to extend results of Wisniewski [11] and some of results of Hsing [4] and Hiisler [5] to the stationary case of multidimensional extreme order statistics and multivariate point processes of exceedances. Identical with Hiisler's [5] assumptions are used, apart from the condition which is a slightly stronger version of the long range dependence condition D^ (see Section 3).…”

“…Let {5'" : n G K} denote the sequence of the d-variate point processes of exceedances obtained on the base £ and previously considered normalization (see Wisniewski [11,12] and the one-dimensional case in Leadbetter et al [6]). This means that…”

“…It is sufficient to use Corollaries 3 and 6 of Wisniewski [11] together with Theorem 3.6 of Hiisler [5].…”

“…Since a multivariate simple point process is Poisson process if and only if its marginals are independent Poisson point processes (see Corollary 1. of Wisniewski [11]), Corollary 5.3 asserts that limit point processes of exceedances are not always those of Poisson. It results from Corollaries 5 and 6 of Wisniewski [11].…”