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

Theodore P. Hill and M. C. Spruill
Proceedings of the American Mathematical Society
Vol. 121, No. 4 (Aug., 1994), pp. 1235-1243
DOI: 10.2307/2161237
Stable URL: http://www.jstor.org/stable/2161237
Page Count: 9
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On the Relationship Between Convergence in Distribution and Convergence of Expected Extremes
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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 $\lim_{n \rightarrow \infty} M_k(X_n) = \alpha_k$ for all k ≥ 1, then there exists a random variable X with Mk(X) = αk for all k ≥ 1 and $X_n \overset{\mathscr{L}}{\longrightarrow} X$ 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.

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