PROPHET INEQUALITIES FOR BOUNDED NEGATIVELY DEPENDENT RANDOM-VARIABLES

It is shown that if X(k) satisfy P(X(k) < a(k)\X1 < a1,..., X(k-1) < a(k-1)) is nondecreasing in a1,..., a(k-1), a negative dependence condition slightly weaker than CDS, and 0 less-than-or-equal-to X(k) less-than-or-equal-to 1, then E[max X(k)] less-than-or-equal-to 2V-V2, where V = sup EX(t), t a stopping rule, holds both for finite and infinite sequences X1, X2,.... Actually, here V can be replaced by the optimal value attainable by threshold rules.