Stop Rule Inequalities for Uniformly Bounded Sequences of Random Variables

If X"0, Xx_ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and if V( X0,X¡,... ) is the supremum, over stop rules /, of EX,, then the set of ordered pairs {(.*, v): x V(X0, Xx,.. .,Xn) and y £(maxyS„X¡) for some X0,..., Xn] is precisely the set C„= {(x,y):x<y<x(\ + n(\ *'/")); 0 « x « l}; and the set of ordered pairs {(x, y): x V(X0, X,,...) and y £(sup„ X„) for some X0, X,,...} is precisely the set X C= UQ. »=i As a special case, if A"0, X,,... is a martingale with EX0 x, then £(max7tí„ X) =c x + nx(\ x'/n) and £(sup„ Xlt) « x x\n x, and both inequalities are sharp.