Stop rule and supremum expectations of i.i.d. random variables: A complete comparison by conjugate duality. (English) Zbl 0598.60044
\(X_ 1,...,X_ n\) 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.
Reviewer: L.Weiss
MSC:
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
| 60E15 | Inequalities; stochastic orderings |
| 62L15 | Optimal stopping in statistics |
| 90C99 | Mathematical programming |
Keywords:
optimal stopping; extremal distributions; inequalities for stochastic processes; conjugate duality; Young’s inequalityReferences:
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