Expectation, conditional expectation and martingales in local fields. (English) Zbl 1127.60005

Evans, Steven N.; Lidman, Tye

Electron. J. Probab. 12, 498-515 (2007).

Summary: 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, the 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.

MSC:

60B99	Probability theory on algebraic and topological structures
60A10	Probabilistic measure theory
60G48	Generalizations of martingales

Keywords:

expectation; conditional expectation; projection; optional sampling; martingale convergence