Expectation, Conditional Expectation and Martingales in Local Fields
Steven N. Evans, University of California at Berkeley
Tye Lidman, University of California at Berkeley
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
L2 the expected value is the orthogonal projection onto the
constants. Previous work has shown that the local field version of
L∞ is the appropriate counterpart of L2, and so the expected
value of a local field-valued random variable is defined to be its
``projection'' in L∞ 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∞ 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.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. You can even send a scanned jpeg or pdf of this copyright form to the managing editor ejpecpme@math.univ-nantes.fr. as an attached file.
If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.