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2007 Expectation, Conditional Expectation and Martingales in Local Fields
Steven Evans, Tye Lidman
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Electron. J. Probab. 12: 498-515 (2007). DOI: 10.1214/EJP.v12-405


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.


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Steven Evans. Tye Lidman. "Expectation, Conditional Expectation and Martingales in Local Fields." Electron. J. Probab. 12 498 - 515, 2007.


Accepted: 18 April 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1127.60005
MathSciNet: MR2299926
Digital Object Identifier: 10.1214/EJP.v12-405

Primary: 60B99
Secondary: 60A10 , 60G48

Keywords: conditional expectation , expectation , Local Field , martingale , martingale convergence , optional sampling , projection

Vol.12 • 2007
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