#### Recommended Citation

Published in *Transactions of the American Mathematical Society*, Volume 278, Issue 1, July 1, 1983, pages 197-207. Copyright © 1983 American Mathematical Society. The definitive version is available at http://www.jstor.org/stable/1999311.

*NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly*.

#### Abstract

If *X*_{0}, *X*_{1}, ... is an arbitrarily-dependent sequence of random variables taking values in [0,1] and if *V*( *X*_{0}, *X*_{1}, ...) is the supremum, over stop rules *t*, of *EX*_{f}, then the set of ordered pairs {(*x , y*): x = *V*( *X*_{0}, *X*_{1}, ..., *X*_{n} and *y* = E(max_{jXj for some X0, ... , Xn} is precisely the set Cn = {(x, y): x < y < x(1 + n(1 - x1/n)); 0 < x 1}; and the set of ordered pairs {(x, y): x = V( X0, X1, ...) and y = E(supn Xn) for some X0, X1, ... is precisely the set C = U Cn As a special case, if X0, X1, ... is a martingale with EX0 = x, then y = E(maxj X < x + nx(1 - x1/n) and E(supn Xn) < x - x ln x, and both inequalities are sharp.}

#### Disciplines

Mathematics

**URL:** https://digitalcommons.calpoly.edu/rgp_rsr/57