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A connection between supermodular ordering and positive/negative association. (English) Zbl 1034.60016
Let \(X_1,X_2,\dots,X_n\) be real-valued random variables, and let \(X_1^*,X_2^*,\dots,X_n^*\) be independent random variables with \(X_i^*=_{\text{st}}X_i\), \(i=1,2,\dots,n\). The authors show that if \(X_1,X_2,\dots,X_n\) are weakly positively [negatively] associated, then \((X_1,X_2,\dots,X_n)\) is larger [smaller] than \((X_1^*,X_2^*,\dots,X_n^*)\) in the supermodular stochastic order. This generalizes some results of Q.-M. Shao [J. Theor. Probab. 13, 343–356 (2000; Zbl 0971.60015)]. The authors also improve a Kolmogorov-type inequality of P. Matula [Stat. Probab. Lett. 15, 209–213 (1992; Zbl 0925.60024)]. Finally, they present a Rosenthal-type inequality for weakly positively associated random variables.

MSC:
60E15 Inequalities; stochastic orderings
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