Additive comparisons of stop rule and supremum expectations of uniformly bounded independent random variables

1981/03/01 by Theodore P. Hill, T. P. Hill, Robert P. Kertz · 4 citations
Mathematics · Decision Sciences · #Probability and Statistical Research #Probability and Risk Models #Auction Theory and Applications

paper · doi:10.1090/s0002-9939-1981-0627697-3

Abstract

Let XI, X2, . . . be independent random variables taking values in [a, b], and let T denote the stop rules for X1, X2, Then E(supn>1 Xn) - sup EXt 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.

Cited by

• A prophet inequality for independent random variables with finite variances
• Prophet inequalities for cost of observation stopping problems
• Stop rule and supremum expectations of i.i.d. random variables: A complete comparison by conjugate duality
• Moment-based minimax stopping functions for sequences of random variables

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