Summary
Convergence of the sample maximum to a nondegenerate random variable, as the sample sizen→∞, implies the convergence in distribution of thek largest sample extremes to ak-dimensional random vectorM k , for all fixedk. If we letk=k(n)→∞,k/n→0, then a question arises in a natural way: how fast cank grow so that asymptotic probability statements are unaffected when sample extremes are replaced byM k . We answer this question for two classes of events-the class of all Lebesgue sets inR k and the class of events of the form\(\left( {x \in R^k :\sum\limits_1^k {x_i } \leqq a} \right)\).
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Weissman, I. On the asymptotic joint distribution of an unbounded number of sample extremes. Probab. Th. Rel. Fields 78, 39–50 (1988). https://doi.org/10.1007/BF00718033
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DOI: https://doi.org/10.1007/BF00718033
Keywords
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Probability Statement
- Joint Distribution