Proceedings of the American Mathematical Society

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



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.

References [Enhancements On Off] (What's this?)

  • [1] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
  • [2] Patrick Billingsley, Probability and measure, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. MR 830424
  • [3] Peter J. Downey, Distribution-free bounds on the expectation of the maximum with scheduling applications, Oper. Res. Lett. 9 (1990), no. 3, 189–201. MR 1059847, 10.1016/0167-6377(90)90018-Z
  • [4] Peter J. Downey and Robert S. Maier, Orderings arising from expected extremes, with an application, Stochastic inequalities (Seattle, WA, 1991) IMS Lecture Notes Monogr. Ser., vol. 22, Inst. Math. Statist., Hayward, CA, 1992, pp. 66–75. MR 1228056, 10.1214/lnms/1215461943
  • [5] Janos Galambos, The asymptotic theory of extreme order statistics, John Wiley & Sons, New York-Chichester-Brisbane, 1978. Wiley Series in Probability and Mathematical Statistics. MR 489334
  • [6] C. Müntz, Über den approximationssatz von Weierstrass, Schwartz-Festschrift, 1914.
  • [7] I. Natanson, Constructive function theory, Vol. II, Fredrick Ungar, New York, 1965.
  • [8] James Pickands III, Moment convergence of sample extremes, Ann. Math. Statist. 39 (1968), 881–889. MR 0224231
  • [9] Moshe Pollak, On equal distributions, Ann. Statist. 1 (1973), 180–182. MR 0331582

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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