Prophet regions for independent random variables with increasing bounds

Abstract

Let X = (X1, …, Xn) be a sequence of independent, integrable[ai, bi]-valued random variables, where a1 ≤ … ≤ an, b1 ≤ … ≤ bn . Considering the class of all such sequences, a complete comparison is made between M(X), the expected gain of a prophet (an observer with complete foresight), and V(X) the maximal expected gain of a gambler (an observer using only non-anticipatory stopping rules). The solution of this problem is a set in , the ‘prophet region’, which is explicitly characterized. This region yields a variety of prophet inequalities, e.g. M(X) ≤ V(X)/2 if bn = 0, bn-1 = -1, an = -2 and M(X) - V(X) ≤ an/2 if an > 0, bn-1 = 2an, bn = 3an .