Optimal Stopping-Related Inequalities for Iid Random Variables when the Future Is Discounted

Comparisons are made between the maximal expected gain of a prophet and the maximal expected reward of an ordinary player observing a sequence of uniformly bounded i.i.d. random variables when the future is discounted. The player uses pure threshold stopping times which are asymptotically optimal. Both finite and infinite sequences of random variables are treated. Also, comparisons between optimal stopping values and expected rewards obtained by using asymptotically optimal pure threshold stopping time are given.