Title

On the Relationship Between Convergence in Distribution and Convergence of Expected Extremes

Author Info

Theodore P. Hill, Georgia Institute of Technology - Main CampusFollow
M. C. Spruill, Georgia Institute of Technology - Main Campus

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_n} is also uniformly integrable. In addition, it is shown using a theorem of Müntz that any subsequence {M_kj (X)}, j ≥ 1, satisfying Σ 1/k_j = ∞ uniquely determines the law of X.