Multidimensional point processes of extreme order statistics
Mateusz Wiśniewski · Demonstratio Mathematica · 1994
Let X n ;n £ N be a sequence of independent random vectors with the common distribution function F. In this paper it is proved that the sequence of multivariate point processes of extreme order statistics, built on the base of the sequence X n ] n 6 N, has the simple limit distribution of Poisson's components.It is also proved that the limit distribution is of Poisson's type if and only if the distribution function F is asymptotically independent (the term of asymptotic independence comes from the extreme value theory).Additionally, this paper contains the conclusions connected with the existence and characterization of the limit distributions of multidimensional order statistics. Notes and definitionsLet E be a complete and separable metric space with Borel c-algebra *B(E).Then the space E' = E x {1,..., d) with the metric g'((x, i), (y,j)) = I * -j I) where g denotes the metric in E, is also complete and separable.Borel <7-algebra ©(£') consists of the sets of the form U?=i F* x {¿}, where F { € »(£).Let Mp(E') denote the set of all boundedly finite measures defined on ©(£') taking values in the set {0} U N U {oo}.Let us denote by <m p (E') the smallest <7-algebra containing the sets of the form {m G M P (E') : m( F') 6 B} for F' € »(£') and B £ »([0, oo]).The d-dimensional point process on E' is defined as (see Daley, Vere-Jones [3], Definition 7.1.XII) a measurable mapping: N:(i2,il,P)^(M p (E'),m p (E')).