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Wald for non-stopping times: the rewards of impatient prophets

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Wald for non-stopping times: the rewards of impatient prophets
2.	Creator	Author's name, affiliation, country	Alexander E Holroyd; Microsoft Research; United States
2.	Creator	Author's name, affiliation, country	Yuval Peres; Microsoft Research; United States
2.	Creator	Author's name, affiliation, country	Jeffrey E Steif; Chalmers University of Technology and Göteborg University; Sweden
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Wald's identity; stopping time; moment condition; prophet inequality
3.	Subject	Subject classification	60G50
4.	Description	Abstract	Let $X_1,X_2,\ldots$ be independent identically distributed nonnegative random variables. Wald's identity states that the random sum $S_T:=X_1+\cdots+X_T$ has expectation $\mathbb{E} T \cdot \mathbb{E} X_1$ provided $T$ is a stopping time. We prove here that for any $1<\alpha\leq 2$, if $T$ is an arbitrary nonnegative random variable, then $S_T$ has finite expectation provided that $X_1$ has finite $\alpha$-moment and $T$ has finite $1/(\alpha-1)$-moment. We also prove a variant in which $T$ is assumed to have a finite exponential moment. These moment conditions are sharp in the sense that for any i.i.d. sequence $X_i$ violating them, there is a $T$ satisfying the given condition for which $S_T$ (and, in fact, $X_T$) has infinite expectation.An interpretation of this is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2014-11-12
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/3609
10.	Identifier	Digital Object Identifier	10.1214/ECP.v19-3609
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 19
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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