Expectation, Conditional Expectation and Martingales in Local Fields
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 realvalued 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 fieldvalued 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 realvalued 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606609
 Bibcode:
 2006math......6609E
 Keywords:

 Mathematics  Probability;
 60A10;
 60B99;
 60G48
 EPrint:
 19 pages