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On the relationship between convergence in distribution and convergence of expected extremes

Authors: Theodore P. Hill and M. C. Spruill
Journal: Proc. Amer. Math. Soc. 121 (1994), 1235-1243
MSC: Primary 60F99; Secondary 60G70
Erratum: Proc. Amer. Math. Soc. 128 (2000), 625-626.
MathSciNet review: 1195722
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Abstract: It is well known that the expected values $ \{ {M_k}(X)\} , k \geq 1$, of the k-maximal order statistics of an integrable random variable X uniquely determine the distribution of X. The main result in this paper is that if $ \{ {X_n}\} , n \geq 1$, is a sequence of integrable random variables with $ {\lim _{n \to \infty }}{M_k}({X_n}) = {\alpha _k}$ for all $ k \geq 1$, then there exists a random variable X with $ {M_k}(X) = {\alpha _k}$ for all $ k \geq 1$ and $ {X_n}\xrightarrow{\mathcal{L}}X$ if and only if $ {\alpha _k} = o(k)$, in which case the collection $ \{ {X_n}\} $ is also uniformly integrable. In addition, it is shown using a theorem of Müntz that any subsequence $ \{ {M_{{k_j}}}(X)\} , j \geq 1$, satisfying $ \sum 1/{k_j} = \infty $ uniquely determines the law of X.

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Keywords: Convergence in distribution, convergence of expected extremes, maximal order statistics, Müntz's theorem, extreme value-theory
Article copyright: © Copyright 1994 American Mathematical Society

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