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.
Highlights
Expectation and conditional expectation of real-valued random variables and the corresponding notion of martingale are fundamental objects of probability theory
In this paper we investigate whether there are analogous notions for random variables with values in a local field – a setting that shares the linear structure which underpins many of the properties of the classical entities
If we identify two random variables as being equal when they are equal almost surely
Summary
Expectation and conditional expectation of real-valued random variables (or, more generally, Banach space-valued random variables) and the corresponding notion of martingale are fundamental objects of probability theory. The appropriate definition of the conditional expectation of X ∈ L∞ given a sub-σ-field G ⊆ F is not, as one might first imagine, the L∞ projection of X onto L∞(G) (:= the subspace of L∞ consisting of G-measurable random variables) Note: We adopt the convention that all equalities and inequalities between random variables should be interpreted as holding P-almost surely
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have