# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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## Stop rule inequalities for uniformly bounded sequences of random variablesHTML articles powered by AMS MathViewer

by Theodore P. Hill and Robert P. Kertz
Trans. Amer. Math. Soc. 278 (1983), 197-207 Request permission

## 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 ${C_n} = \{ (x,y):x \leqslant y \leqslant x ( {1 + n (1 - {x^{1/n}})} );0 \leqslant x \leqslant 1\} ;$ 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 $C = \bigcup \limits _{n = 1}^\infty {{C_n}} .$ 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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