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On the Relationship Between Convergence in Distribution and Convergence of Expected Extremes
Proceedings of the American Mathematical Society
  • Theodore P. Hill, Georgia Institute of Technology - Main Campus
  • M. C. Spruill, Georgia Institute of Technology - Main Campus
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/