Course in Probability Theory

9 .1 BASIC PROPERTIES OF CONDITIONAL EXPECTATION 1 3 17(vii) If J X, I < Y where c`(Y) < oo and X„ -+ X, then ((X„ I l')~` (X 1 10 .To illustrate, (iii) is proved by observing that for each A E < :f~f(Xi I ~)dJT=f X i d2 ~`'(X2 I )}, we have 7'(A) = 0 . The inequality(iv) may be proved by (ii), (iii), and the equation X = X+ - X- . To prove(v), let the limit of f (X„ 1 ~) be Z, which exists a .e . by (iii) . Then for eachA E {, we have by the monotone convergence theorem :JZ dT = lim ' ~ (X„ I ~) dT = lim X„ dT = f X dT .A n f A " f A AThus Z satisfies the defining relation for o (X 1 6'), and it belongs to 6' withthe F (X„ 16')'s, hence Z = e(X 16') .To appreciate the caution that is necessary in handling conditional expectations,let us consider the Cauchy-Schwarz inequality :(IXYI 16)2 < e(X 2 I -6')A(Y 2 I ') .If we try to extend one of the usual proofs based on the positiveness of thequadratic form in A : (`((X + AY) 2 16'), the question arises that for each ),the quantity is defined only up to a null set N;,, and the union of these overall A cannot be ignored without comment . The reader is advised to think thisdifficulty through to its logical end, and then get out of it by restricting theA's to the rationals . Here is another way out : start from the following trivialinequality :IXIIYI X 2 y2up- 2a2 + 2fl 2 'where a = (X 2 1 6)12 , ,B = C~(Y 2 16) 1 / 2 , and a,B>0 ; apply the operation{- 16} using (ii) and (iii) above to obtainIXYIca,81 X 2 1 Y ~ 2< 2 c( a2 l~ + ZenNow use Theorem 9 .1 .3 to infer that this can be reduced toufi2 2{IXYIIf}< - a2 +2 2=1,the desired inequality .The following theorem is a generalization of Jensen's inequality in Sec . 3 .2 .