Limit theory for multivariate sample extremes


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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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).

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