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Expectation, Conditional Expectation and Martingales in Local Fields

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Expectation, Conditional Expectation and Martingales in Local Fields
2.	Creator	Author's name, affiliation, country	Steven N. Evans; University of California at Berkeley
2.	Creator	Author's name, affiliation, country	Tye Lidman; University of California at Berkeley
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	local field; expectation; conditional expectation; projection; martingale; martingale convergence; optional sampling
3.	Subject	Subject classification	60B99; 60A10; 60G48
4.	Description	Abstract	We investigate a possible definition of expectation and conditional expectation for random variables with values in a local field such as the $p$-adic numbers. We define the expectation by analogy with the observation that for real-valued random variables in $L^2$ the expected value is the orthogonal projection onto the constants. Previous work has shown that the local field version of $L^\infty$ is the appropriate counterpart of $L^2$, and so the expected value of a local field-valued random variable is defined to be its ``projection'' in $L^\infty$ onto the constants.  Unlike the real case, the resulting projection is not typically a single constant, but rather a ball in the metric on the local field.  However, many properties of this expectation operation and the corresponding conditional expectation mirror those familiar from the real-valued case; for example, conditional expectation is, in a suitable sense, a contraction on $L^\infty$ and the tower property holds. We also define the corresponding notion of martingale, show that several standard examples of martingales (for example, sums or products of suitable independent random variables or ``harmonic'' functions composed with Markov chains) have local field analogues, and obtain versions of the optional sampling and martingale convergence theorems.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	NSF grant DMS-0405778; NSF VIGRE grant DMS-0130526
7.	Date	(YYYY-MM-DD)	2007-04-18
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/405
10.	Identifier	Digital Object Identifier	10.1214/EJP.v12-405
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 12
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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