Some special properties of conditional expectation

Abstract

In this paper we shall discuss some special properties of conditional expectation. In Section I we prove that the condition given in [7] is equivalent, in some sense, to the assertion of the Fatou Lemma.The conditional expectation can be defined as an orthogonalprojection. We show that the almost sure convergence of a sequence of conditional expectations of random variables {E~-Xn, n __> 1} does not follow from the almost sure convergence of the sequence {Xn, n __> 1}, and conversely. Some suitable examples are given in Section II.In the problem of optimal stopping one considers the value sup EXt. In rET [4] Sudderth proved that E lim sup X,, = lira sup E X~-vET if IXnl < Z, where Z is an integrable r.v. In Section III we generalize the relation obtained by Sudderth to the conditional case. Moreover, we give an equivalent condition to this equality. I. The conditional Fatou LemmaLet (~t, A, P) be a probability space, {iTn, n > 1} an increasing sequence of sub-a-fields contained in A and {Xn, n > 1} a sequence of r.v. such that Xn is S'n-measurable for every n.The definition of conditional expectation (E'rX) with respect to the sub-a-field iT C A for X > 0 can be found in [3]. If min(EYX +, ETX -) < oc a.s.where X + = max(X, 0), X-= max(-X, 0), then we define the conditional expectation as follows: E-rX = E-rX + _ E-rX -.In case Er]X] < oc a.s. we write X E L~=.