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Prophet regions for discounted, uniformly bounded random variables

Summary: Prophet regions for discounted,
uniformly bounded random variables
Pieter C. Allaart
University of North Texas
September 25, 2005
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 Bj-1Xj,
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 -
)]} is the prophet region for undiscounted random variables given by Hill and
Kertz (Trans. Amer. Math. Soc. 278, 197-207 (1983)). As a special case, it is shown
that the maximum possible difference E(maxjn j-1
Xj)-supt E(t-1


Source: Allaart, Pieter - Department of Mathematics, University of North Texas


Collections: Mathematics