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Stop Rule Inequalities for Uniformly Bounded Sequences of Random Variables
Transactions of the American Mathematical Society
  • Theodore P. Hill, Georgia Institute of Technology - Main Campus
  • Robert P. Kertz, Georgia Institute of Technology - Main Campus
Publication Date
7-1-1983
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 EXf, then the set of ordered pairs {(x , y): x = V( X0, X1, ..., Xn and y = E(maxjXj 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
Citation Information
Theodore P. Hill and Robert P. Kertz. "Stop Rule Inequalities for Uniformly Bounded Sequences of Random Variables" Transactions of the American Mathematical Society Vol. 278 Iss. 1 (1983) p. 197 - 207
Available at: http://works.bepress.com/tphill/1/