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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the relationship between convergence in distribution and convergence of expected extremes
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by Theodore P. Hill and M. C. Spruill PDF
Proc. Amer. Math. Soc. 121 (1994), 1235-1243 Request permission

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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • 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
  • MathSciNet review: 1195722