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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Stop rule inequalities for uniformly bounded sequences of random variables


Authors: Theodore P. Hill and Robert P. Kertz
Journal: Trans. Amer. Math. Soc. 278 (1983), 197-207
MSC: Primary 60G40; Secondary 60G42, 62L15
MathSciNet review: 697070
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Abstract: If $ {X_{0}},{X_1},\ldots$ is an arbitrarily-dependent sequence of random variables taking values in $ [0,1]$ and if $ V({X_0},{X_1},\ldots)$ is the supremum, over stop rules $ t$, of $ E{X_t}$, then the set of ordered pairs $ \{ (x,y):x = V({X_0},{X_1},\ldots,{X_n})$ and $ y = E({\max_{j\, \leqslant \,n}}{X_j})$ for some $ {X_0},\ldots,{X_n}\} $ is precisely the set

$\displaystyle {C_n} = \{ (x,y):x \leqslant y \leqslant x\,( {1 + n\,(1 - {x^{1/n}})} );0 \leqslant x \leqslant 1\} ;$

and the set of ordered pairs $ \{ (x,y):x = V({X_{0}},{X_1},\ldots)$ and $ y = E({\sup_n}\;{X_n})$ for some $ {X_0},{X_1},\ldots\} $ is precisely the set

$\displaystyle C = \bigcup\limits_{n = 1}^\infty {{C_n}} .$

As a special case, if $ {X_0},{X_1},\ldots$ is a martingale with $ E{X_0} = x$, then $ E({\max_{j \leqslant n}} X) \leqslant x + nx(1 - {x^{1\,/\,n}})$ and $ E({\sup_n}\;{X_n}) \leqslant x - x\ln \;x$, and both inequalities are sharp.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0697070-7
PII: S 0002-9947(1983)0697070-7
Keywords: Optimal stopping, extremal distributions, inequalities for stochastic processes, prophet inequalities, martingales, Markov processes
Article copyright: © Copyright 1983 American Mathematical Society