Prophet regions for independent random variables with increasing bounds: Prophet regions for independent random variables

Let X = (X 1 , ..., X n ) be a sequence of independent, integrable[a i , b i ]-valued random variables, where a 1 ≤ ... ≤ a n , b 1 ≤ ... ≤ b n . 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 foresig...

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Bibliographic Details
Published in:Sequential analysis 1998-01, Vol.17 (2), p.195-204
Main Authors: Uwe Schmid, Uwe Saint-Mont
Format: Article
Language:English
Subjects:
Inequalities for stochastic processes
Optimal stopping
Prophet inequalities
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Summary:Let X = (X 1 , ..., X n ) be a sequence of independent, integrable[a i , b i ]-valued random variables, where a 1 ≤ ... ≤ a n , b 1 ≤ ... ≤ b n . 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 b n = 0, b n-1 = -1, a n = -2 and M(X) - V(X) ≤ a n /2 if a n > 0, b n-1 = 2a n , b n = 3a n .
ISSN:0747-4946
1532-4176
DOI:10.1080/07474949808836407