 
Summary: Prophet inequalities for i.i.d. random variables
with random arrival times.
Pieter C. Allaart
University of North Texas
October 3, 2005
Abstract
Suppose X1, X2, . . . are i.i.d. nonnegative random variables with finite expec
tation, and for each k, Xk is observed at the kth arrival time Sk of a Poisson
process with unit rate which is independent of the sequence {Xk}. Let t > 0 be
a finite time horizon. Several comparisons are made between the expected max
imum M(t) := E[maxk1 Xk I(Sk t)] and the optimal stopping value V (t) :=
supT E[X I(S t)], where T is the set of all INvalued random variables such
that { = i} is measurable with respect to the algebra generated by X1, . . . , Xi and
S1, . . . , Si. For instance, it is shown that M(t)/V (t) 1 + 0, where 0
.
= 0.34149
is the unique value of such that
1
0
(y  y ln y + )1
