Title

Stop Rule Inequalities for Uniformly Bounded Sequences of Random Variables

Author Info

Theodore P. Hill, Georgia Institute of Technology - Main CampusFollow
Robert P. Kertz, Georgia Institute of Technology - Main Campus

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 X₀, X₁, ... is an arbitrarily-dependent sequence of random variables taking values in [0,1] and if V( X₀, X₁, ...) is the supremum, over stop rules t, of EX_f, then the set of ordered pairs {(x , y): x = V( X₀, X₁, ..., X_n and y = E(max_{jXj for some X0, ... , Xn} is precisely the set Cn = {(x, y): x < y < x(1 + n(1 - x^1/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 - x^1/n) and E(supn Xn) < x - x ln x, and both inequalities are sharp.}

Disciplines

Mathematics