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 $\{ {M_k}(X)\} , k \geq 1$, 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*.

- Patrick Billingsley,
*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396** - 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** - 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**, DOI https://doi.org/10.1016/0167-6377%2890%2990018-Z - 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**, DOI https://doi.org/10.1214/lnms/1215461943 - 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**
C. Müntz, - James Pickands III,
*Moment convergence of sample extremes*, Ann. Math. Statist.**39**(1968), 881–889. MR**224231**, DOI https://doi.org/10.1214/aoms/1177698320 - Moshe Pollak,
*On equal distributions*, Ann. Statist.**1**(1973), 180–182. MR**331582**

*Über den approximationssatz von Weierstrass*, Schwartz-Festschrift, 1914. I. Natanson,

*Constructive function theory*, Vol. II, Fredrick Ungar, New York, 1965.

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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