Stop rule inequalities for uniformly bounded sequences of random variables

Abstract:

If ${X_{0}},{X_1},\ldots$ is an arbitrarily-dependent sequence of random variables taking values in $[0,1]$ and if $V({X_0},{X_1},\ldots )$ is the supremum, over stop rules $t$, of $E{X_t}$, then the set of ordered pairs $\{ (x,y):x = V({X_0},{X_1},\ldots ,{X_n})$ and $y = E({\max _{j \leqslant n}}{X_j})$ for some ${X_0},\ldots ,{X_n}\}$ is precisely the set \[ {C_n} = \{ (x,y):x \leqslant y \leqslant x ( {1 + n (1 - {x^{1/n}})} );0 \leqslant x \leqslant 1\} ;\] and the set of ordered pairs $\{ (x,y):x = V({X_{0}},{X_1},\ldots )$ and $y = E({\sup _n}\;{X_n})$ for some ${X_0},{X_1},\ldots \}$ is precisely the set \[ C = \bigcup \limits _{n = 1}^\infty {{C_n}} .\] As a special case, if ${X_0},{X_1},\ldots$ is a martingale with $E{X_0} = x$, then $E({\max _{j \leqslant n}} X) \leqslant x + nx(1 - {x^{1 / n}})$ and $E({\sup _n}\;{X_n}) \leqslant x - x\ln \;x$, and both inequalities are sharp.