Prophet inequalities for averages of independent non-negative random variables
Let \(X_ 1,X_ 2,..\). be independent nonnegative random variables, \(A_ n=(X_ 1+...+X_ n)/n\) and \(\Sigma_ n\) the set of stopping times \(\tau\) for \(X_ 1,X_ 2,...\). The author proves \[ E(\sup_{j\leq n}A_ j)\leq (2-1/n)\sup \{E(A_{\tau}): \tau \in \Sigma_ n\}. \] The constant (2-1/n) is sharp. (The reviewer and \textit{L. Sucheston} [Bull. Am. Math. Soc. 83, 745-747 (1977; Zbl 0336.60032)] had obtained the inequality with the weaker constant \(2(1+\sqrt{3})\), and the problem of obtaining the best possible constant had challenged quite a few mathematicians).
scientific article; zbMATH DE number 3976008
scientific article