#### Recommended Citation

Published in *Proceedings of the American Mathematical Society*, Volume 121, Issue 4, August 1, 1994, pages 1235-1243. Copyright © 1994 American Mathematical Society. The definitive version is available at http://www.jstor.org/stable/2161237.

*NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly*.

#### Abstract

It is well known that the expected values {*M _{k}*(

*X*)},

*k*≤ 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*≥ 1, is a sequence of integrable random variables with lim

_{n -> ∞}

*M*(

_{k}*X*

_{n}) = α

_{k}for all

*k*≥ 1, then there exists a random variable

*X*with

*M*(

_{k}*X*) = α

_{k}for all

*k*≥ 1 and

*X*->

_{n}*L*->

*X*if and only if α

_{k}= o(

*k*), in which case the collection {

*X*} is also uniformly integrable. In addition, it is shown using a theorem of Müntz that any subsequence {

_{n}*M*(

_{kj}*X*)},

*j*≥ 1, satisfying Σ 1/

*k*= ∞ uniquely determines the law of

_{j}*X*.

**URL:** https://digitalcommons.calpoly.edu/rgp_rsr/32