 
Summary: Prophet regions for discounted,
uniformly bounded random variables
Pieter C. Allaart
University of North Texas
September 25, 2005
Abstract
Let X1, X2, . . . be any sequence of [0, 1]valued random variables. A complete
comparison is made between the expected maximum E(maxjn Yj) and the stop rule
supremum supt E Yt for two types of discounted sequences: (i) Yj = bjXj, where {bj} is
a nonincreasing sequence of positive numbers with b1 = 1; and (ii) Yj = B1 · · · Bj1Xj,
where B1, B2, . . . are independent [0, 1]valued random variables that are independent
of the Xj, having a common mean . For instance, it is shown that the set of points
{(x, y) : x = supt E Yt and y = E(maxjn Yj) for some sequence X1, . . . , Xn and
Yj = bjXj} is precisely the convex closure of the union of the sets {(bjx, bjy) : (x, y)
Cj}, j = 1, . . . , n, where Cj = {(x, y) : 0 x 1, x y x[1 + (j  1)(1 
x1/(j1)
)]} is the prophet region for undiscounted random variables given by Hill and
Kertz (Trans. Amer. Math. Soc. 278, 197207 (1983)). As a special case, it is shown
that the maximum possible difference E(maxjn j1
Xj)supt E(t1
