Prophet inequalities for bounded negatively dependent random variables

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It is shown that if X k satisfy P( X k < a k ∣ X 1 < a 1,…, X k-1 < a k-1 ) is nondecreasing in a 1,…, a k-1 , a negative dependence condition slightly weaker than CDS, and 0 ⩽ X k ⩽ 1, then E[max X k ] ⩽ 2 V – V 2, where V = sup EX t , t, a stopping rule, holds both for finite and infinite sequences X 1, X 2,…. Actually, here V can be replaced by the optimal value attainable by threshold rules.