# On the asymptotic joint distribution of an unbounded number of sample extremes

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## Summary

Convergence of the sample maximum to a nondegenerate random variable, as the sample size*n*→∞, implies the convergence in distribution of the*k* largest sample extremes to a*k*-dimensional random vector**M**_{ k }, for all fixed*k*. If we let*k=k(n)*→∞,*k/n*→0, then a question arises in a natural way: how fast can*k* grow so that asymptotic probability statements are unaffected when sample extremes are replaced by**M**_{ k }. We answer this question for two classes of events-the class of all Lebesgue sets in*R*^{ k } and the class of events of the form\(\left( {x \in R^k :\sum\limits_1^k {x_i } \leqq a} \right)\).

## Keywords

Stochastic Process Probability Theory Mathematical Biology Probability Statement Joint Distribution
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