Article

Stop Rule Inequalities for Uniformly Bounded Sequences of Random Variables

Transactions of the American Mathematical Society
Publication Date

7-1-1983
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(maxj*X*j for some *X*0, ... , *X*n} is precisely the set *C*n = {(*x, y*): *x* < *y* < *x*(1 + *n*(1 - x1/n)); 0 < *x* 1}; and the set of ordered pairs {(*x, y*): *x* = *V*( *X*0, *X*1, ...) and *y* = *E*(supn Xn) for some *X*0, *X*1, ... is precisely the set *C* = U *C*n As a special case, if *X*0, *X*1, ... is a martingale with *EX*0 = 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/