Summary
If X 1, X 2, ..., are i.i.d. random variables and Y n =Max(X 1, ..., X n ); if for some sequences A n , Bn, n=1, 2, ..., E n (t)=AnY[nt]+Bn is such that E n (1) weakly converges to a non degenerate limit distribution, then we prove that it is possible to construct a sequence of replicates of extremal processes E (n)(t) on the same probability space, such that d(E n (.), E (n)(.))→0 a.s., with the Levy metric. We give the rates of consistency of the approximations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barndorff-Nielsen, O.: On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables. Matl2 Scand. 9, 383–394 (1961)
Deheuvels, P.: Valeurs extremales d'échantillons croissants d'une variable aléatoire réelle. Ann. Inst. l2 Poincaré Ser. B10, 89–114 (1974)
Deheuvels, P.: Majoration et minoration presque sûre optimale des éléments de la statistique ordonnée d'un échantillon croissant de variables aléatoires indépendantes. Rend. Acad. Nazionale dei Lincei Ser. VIII Vol. LVI 707–719 (1974)
Galambos, J.: The asymptotic theory of extreme order statistics. New York: Wiley 1978
Hebna-Grabowska, H., Szynal, D.: An almost sure invariance principale for the partial sums of infima of independent random variables. Ann. Probability 7, 1036–1045 (1979)
Parzen, E.: Stochastic processes. San Francisco: Holden Day 1962
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Deheuvels, P. The strong approximation of extremal processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 58, 1–6 (1981). https://doi.org/10.1007/BF00536191
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00536191