Title

Additive Comparisons of Stop Rule and Supremum Expectations of Uniformly Bounded Independent Random Variables

Author Info

Theodore P. Hill, Georgia Institute of Technology - Main CampusFollow
Robert P. Kertz, Georgia Institute of Technology - Main Campus

Recommended Citation

Published in Proceedings of the American Mathematical Society, Volume 83, Issue 3, November 1, 1981, pages 582-585.

Copyright © 1981 The American Mathematical Society.

NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.

The definitive version is available at https://doi.org/10.1090/S0002-9939-1981-0627697-3.

Abstract

Let X_I, X₂, . . . be independent random variables taking values in [a, b], and let T denote the stop rules for X₁, X₂, Then E(sup_n>1 X_n) - sup{ EX_t t ≡ T} < (1/4)(b - a), and this bound is best possible. Probabilistically, this says that if a prophet (player with complete foresight) makes a side payment of (b - a)/8 to a gambler (player using nonanticipating stop rules), the game becomes at least fair for the gambler.

Disciplines

Mathematics