Prophet Inequalities for Averages of Independent Non-Negative Random Variables

The main purpose of this paper is to prove the following theorem, which sharpens results of Krengel and Sucheston [11, 12] in which the weaker constant 2(1 +-(3) was obtained. (Here EX is the expected value of the ran­ dom variable X, and ~ and T are the sets of stop rules ~ n, and of a.s. finite stop rules, respectively.) Theorem 1.1. If Xl' ... , X n are independent non-negative random variables, then and this bound is sharp for all n ~ 1. By passing to limits, one easily obtains the following corollary. (2) and this constant "2" is sharp. Inequalities such as (1) and (2), which compare the expected supremum of a sequence of random variables with the supremum over stop rules of the expected value at the time of stopping, were first discovered by Krengel and Sucheston [11, 12] and have been called "prophet inequalities" because of the probabilistic interpretation of E(sup Zn) as the expected value of the sequence n {Zn} to a "prophet", or player with complete foresight. Such inequalities have been studied for various processes {Zn} including: {Zn} independent and non­