Article

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

Proceedings of the American Mathematical Society
Publication Date

8-1-1994
Abstract

It is well known that the expected values {Mk(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 {Xn}, n ≥ 1, is a sequence of integrable random variables with limn -> ∞ Mk(Xn) = αk for all k ≥ 1, then there exists a random variable X with Mk(X) = αk for all k ≥ 1 and Xn ->L->X if and only if αk = o(k), in which case the collection {Xn} is also uniformly integrable. In addition, it is shown using a theorem of Müntz that any subsequence {Mkj (X)}, j ≥ 1, satisfying Σ 1/kj = ∞ uniquely determines the law of X.
Citation Information

Theodore P. Hill and M. C. Spruill. "On the Relationship Between Convergence in Distribution and Convergence of Expected Extremes" *Proceedings of the American Mathematical Society*Vol. 121 Iss. 4 (1994) p. 1235 - 1243

Available at: http://works.bepress.com/tphill/48/