Mathematical Statistics and Stochastic Processes

Chapter 3

Conditional Expectation

3.1. Definition

Let (X, Y) be a pair of random variables defined on the probability space in which only X is observed. We wish to know what information X carries about Y: this is the filtering problem defined in Chapter 1.

This problem may be formalized in the following way: supposing Y to be real and square integrable, construct a real random variable of the form r(X) that gives the best possible approximation of Y with respect to the quadratic error, i.e. E[(Y − r(X))2] being minimal.

If we identify the random variables with their P-equivalence classes, we deduce that r(X) exists and is unique, since it is the orthogonal projection (in the Hilbert space L2 (P)) of Y on the closed vector subspace L2 (P) constituted by the real random variables of the form h(X) and such that E[(h(X))2] < +∞.

From Doob’s lemma, the real random variables of the form h(X) are those that are measurable with respect to the σ-algebra generated by X. We say that r(X) is the conditional expectation of Y with respect to the σ-algebra generated by X (or with respect to X), and that r is the regression of Y on X. We write:

The above equation leads us to the following definition.

DEFINITION 3.1.– Let be a probability space and let be a sub-σ-algebra of . We call the orthogonal ...