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

Mathematics

Included in

Mathematics Commons

Share

COinS
 

URL: http://digitalcommons.calpoly.edu/rgp_rsr/57