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- John Elton, Theodore P Hill, Robert P Kertz
- 2010

Suppose fix,... ,fin are nonatomic probability measures on the same measurable space (S, S). Then there exists a measurable partition isi}"=i of 5 such that Pi(Si) > (n + 1 M)'1 for a11 i l,...,n, where M is the total mass of V?=i ßi (tne smallest measure majorizing each m). This inequality is the best possible for the functional M, and sharpens and… (More)

This paper surveys the origin and development of what has come to be known as "prophet inequalities" in optimal stopping theory. Included is a review of all published work to date on these problems, in cl uding extensions and variations, descriptions and examples of the main proof techniques, and a list of a number of basic open problems.

If X"0, Xx_ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and if V X C= UQ. »=i As a special case, if A"0, X,,... is a martingale with EX0 x, then £(max7tí " X) =c x + nx(\ x'/n) and £(sup " Xlt) « x x\n x, and both inequalities are sharp. 1. Introduction. The subject of this paper is comparisons between the expected… (More)

- T P, Hill I, Robert P Kertz
- 2006

Implicitly defined (and easily approximated) universal constants 1.1 < an random variables and if Tn is the set of stop rules for Xl, "', Xn, then E(max{Xl , • • • ,Xn}) ~ an sup {EX, : tE Tn}, and the bound an is best possible. Similar universal constants 0 < bn < Y. are found so that if the {Xi} are i.i.d. random variables taking values only in [a, b),… (More)

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