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

DOI:
https://doi.org/10.1090/S0002-9939-1994-1195722-9

Erratum:
Proc. Amer. Math. Soc. **128** (2000), 625-626.

MathSciNet review:
1195722

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Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that the expected values , 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 , is a sequence of integrable random variables with for all , then there exists a random variable *X* with for all and if and only if , in which case the collection is also uniformly integrable. In addition, it is shown using a theorem of Müntz that any subsequence , satisfying uniquely determines the law of *X*.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1195722-9

Keywords:
Convergence in distribution,
convergence of expected extremes,
maximal order statistics,
Müntz's theorem,
extreme value-theory

Article copyright:
© Copyright 1994
American Mathematical Society