Stop rule and supremum expectations of i.i.d. random variables: A complete comparison by conjugate duality
\(X_ 1,...,X_ n\) are independent and identically distributed random variables taking values in the closed interval [0,1]. Define \(V(X_ 1,...,X_ n)\) as \(\sup \{EX_ t:\) t is a stop rule for \(X_ 1,...,X_ n\}\). For x in the closed interval [0,1], a nonnegative strictly increasing strictly concave function \(G_ n(x)\) is constructed such that the set of ordered pairs \(\{(x,y):\) \(x=V(X_ 1,...,X_ n)\) and \(y=E(\max_{j\leq n}X_ j)\) for some \(X_ 1,...,X_ n\}\) is precisely the set \(\{(x,y):\) \(x\leq y\leq G_ n(x)\); \(0\leq x\leq 1\}\). Various inequalities are derived from this result.
scientific article; zbMATH DE number 3965100