Limit theory for multivariate sample extremes

Summary

Let \(\{ (X_n^{(1)} ),...,X_n^{(k)} ,{\text{ }}n\} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 1\} \) be k-dimensional iid random vectors. Necessary and sufficient conditions are found for the weak convergence of the maxima \(\left\{ {\mathop {\text{V}}\limits_{j{\text{ = 1}}}^n X_j^{(1)} ,...,\mathop {\text{V}}\limits_{j{\text{ = 1}}}^n X_j^{(k)} } \right\}\) suitably normed to a non-degenerate limit df.

The class of such limits is specified and conditions stated for the limit joint df to be a product of marginal df's. Some results are presented concerning extremal processes generated by multivariate df's.

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References

  1. 1.

    Balkema, A., Resnick, S.I.: Max-infinite divisibility. J. appl. Probab. 14, 309–319 (1977)

    Google Scholar 

  2. 2.

    Breiman, L.: Probability. Reading, Mass.: Addison Wesley 1969

    Google Scholar 

  3. 3.

    de Haan, L.: On Regular Variation and its Application to the Weak Convergence of Sample Extremes. MC Tract 32, Mathematisch Centrum, Amsterdam (1970)

    Google Scholar 

  4. 4.

    de Haan, L.: A form of regular variation and its application to the domain of attraction of the double exponential distribution. Z. Wahrscheinlichkeitstheorie verw. Gebiete 17, 241–258 (1971)

    Google Scholar 

  5. 5.

    de Haan, L.: Weak limits of sample range. J. appl. Probab. 11, 836–841 (1974)

    Google Scholar 

  6. 6.

    Dwass, M.: Extremal Processes. Ann. math. Statist. 35, 1718–1725 (1964)

    Google Scholar 

  7. 7.

    Galambos, J.: Order statistics from samples from multivariate distributions. J. Amer. statist. Assoc. 70, 674–680 (1975)

    Google Scholar 

  8. 8.

    Geffroy, J.: Contributions à la theorie des valeurs extremes. Publ. Inst. Statist. Univ. Paris, 7, 8, 37–185 (1958)

    Google Scholar 

  9. 9.

    Gnedenko, B.V.: Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 423–453 (1943)

    Google Scholar 

  10. 10.

    Lévy, P.: Théorie de l'addition des variables aléatoires. Paris: Gauthier-Villars, 1937

    Google Scholar 

  11. 11.

    Pickands, J.: Multivariate extreme value distributions. Preprint, University of Pennsylvania (1976)

  12. 12.

    Resnick, S.I., Rubinovitch, M.: On the structure of extremal processes. Advances appl. Probab. 5, 287–307 (1973)

    Google Scholar 

  13. 13.

    Rvačeva, E.L.: On the domains of attraction of multidimensional distributions. Select. Translat. math. Statist. Probab. 2, 183–207 (1962)

    Google Scholar 

  14. 14.

    Sibuya, M: Bivariate extreme statistics. Ann. Inst. statist. Math. Tokyo, 11, 195–210 (1960)

    Google Scholar 

  15. 15.

    Tiago de Oliveira, J.: Extremal distributions. Faculdade de Ciencias de Lisboa, No. 39 (1959)

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Research supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.) and by CSIRO, Division of Mathematics and Statistics, Canberra while on leave from Erasmus University, Rotterdam

Supported provided by National Science Foundation Grant OIP75-14513 while on leave from Stanford University. The hospitality of CSIRO, Division of Mathematics and Statistics, Canberra and the Department of Statistics, SGS, Australian National University is gratefully acknowledged

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de Haan, L., Resnick, S.I. Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheorie verw Gebiete 40, 317–337 (1977). https://doi.org/10.1007/BF00533086

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Keywords

  • Stochastic Process
  • Probability Theory
  • Random Vector
  • Mathematical Biology
  • Weak Convergence