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*(*X*n) = 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/