Prophet regions for independent random variables with simultaneous costs and discountings

Abstract

Let Y = (Y1,Y2,…) be a sequence of independent [l_i,r_i]-valued random variables, where (l_i), (r_i), (d_i):=(r_i-l_i) are non-increasing sequences, and all d_i>0. Considering the class of all such sequences, a complete comparison is made between M(Y), the expected gain of a prophet (an observer with complete foresight), and V(Y), the maximal expected gain of a gambler (unsing only non-anticipatory stopping times). The solution of this problem is a set in , the so-called prophet region, which is determined for finite and infinite sequences. These regions generalize earlier results [4], [6], [1] in a natural manner. Especially, they yield a variety of prophet inequalities, e.g. all results derived in [5], [6] and [1].

^∗Research was in part supported by the Edmund Landau Center for Research in Mathematical Analysis, Jerusalem, Israel. Former surname of the author is ‘Schmid’

^∗Research was in part supported by the Edmund Landau Center for Research in Mathematical Analysis, Jerusalem, Israel. Former surname of the author is ‘Schmid’

Notes

^∗Research was in part supported by the Edmund Landau Center for Research in Mathematical Analysis, Jerusalem, Israel. Former surname of the author is ‘Schmid’