Robert P. Kertz

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ACKNOWLEDGEMENTS There have been many individuals in my life without whom the completion of this work would not have been possible. It is my intension to briefly recognize their contribution to my personal and professional life. I would like to thank my advisor, Dr. Christian Houdré, for his support and orientation through all my studies at Georgia Tech. In(More)
Let XI, X2, . . . be independent random variables taking values in [a, b], and let T denote the stop rules for X1, X2, Then E(sup,, X,,) sup{EX,: t E T) < (1/4)(b a), and this bound is best possible. Probabilistically, this says that if a prophet (player with complete foresight) makes a side payment of (b a)/8 to a gambler (player using nonanticipating stop(More)
Implicitly defined (and easily approximated) universal constants 1.1 < an < 1.6, n = 2, 3, ... , are found so that if XI, X 2 , ••• are i.i.d. non-negative 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(More)
If X"0, Xx_ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and if V( X0,X¡,... ) is the supremum, over stop rules /, of EX,, then the set of ordered pairs {(.*, v): x V(X0, Xx,.. .,Xn) and y £(maxyS„X¡) for some X0,..., Xn] is precisely the set C„= {(x,y):x<y<x(\ + n(\ *'/")); 0 « x « l}; and the set of ordered pairs {(x,(More)
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)
ACKNOWLEDGEMENTS During the preparation of this thesis, there have been many people whose prayer, support, and encouragement have contributed to its accomplishment. First of all, I would like to thank my advisor, Professor Shijie Deng. His thoughtful comments, helpful suggestions, and consistent guidance made the preparation of this thesis an invaluable(More)
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