Prediction and Conditional Expectation

Abstract

Let (Ω, F, P) be a probability space, and let L² be the vector space of random variables Zwith E[Z²<∞. Let X∈L² be an unobservable random variable, whose value we wish to predict from observation of other random variables Y¹,..., Yⁿ. (For example, Xmay be the value of a stochastic process at some time in the future, or a spatial process over a region where it cannot be observed). In order to use knowledge of Y¹,..., Yⁿ to predict X, the predictor must be function of Y¹,..., Yⁿ, and, in particular, is a random variable