Robert P. Kertz

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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)
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)
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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