Prophet-type inequalities for multi-choice optimal stopping
Let \(X_ 1,X_ 2,..\). be independent random variables taking values in [0,1], and let \(v_ r=\sup E(X_{\tau_ 1}+X_{\tau_ 2}+...+X_{\tau_ r})\), where the supremum extends over all stopping times \(\tau_ 1<\tau_ 2<...<\tau_ r\). The author proves inequalities of the form E(sup \(X_ n)\leq F_ r(v_ r).\) He gives an explicit formula for the best possible function \(F_ r\). He also derives best possible constants \(1\leq C_ r\leq 2\) with E(sup \(X_ n)\leq C_ rv_ r\). It is known that \(C_ 1=2\). The constants \(C_ r\) decrease and tend to one rapidly. E.g. \(C_ 2=1.1716\).
scientific article; zbMATH DE number 4007366