Course in Probability Theory

3 24 1 CONDITIONING . MARKOV PROPERTY . MARTINGALEand X2 are independent, what is the effect of condi-Next we ask : if X 1tioning X 1 + X 2 by X 1 ?Theorem 9 .2 .2 . Let X1 and X2 be independent r .v .'s with p .m .'s µl and µ2 ;then for each B E %3 1 :(4) ;/'{X 1 + X2 E B I X 1 } = µ2(B - X 1 ) a .e .More generally, if {X,,, n > 1) is a sequence of independent r.v .'s with p .m .'s{µn , n > 1), and Sn = 2~=1 X j , then for each B E Al(5) A{S n E B I S1, . . . , Sn-1 } _ pn (B - Sn-1) = G1'{Sn E B I Sn-1 } a .e .PROOF . To prove (4), since its right member belongs to :7 {X 1 }, itis sufficient to verify that it satisfies the defining relation for its leftmember. Let A E Z {X1 }, then A = X1 1 (A) for some A E A 1 . It follows fromTheorem 3 .2 .2 thatµ2(B - X1)d9l = [2(B µ-xi)µl(dxl) .fA AWriting A = Al X A2 and applying Fubini's theorem to the right side above,then using Theorem 3 .2.3, we obtainfµl (dxl)A fi+x2EBA2(dx2) = ff µ(dxl, dx2)x, EAx1 +x2EB= dJ' = ~{X1 E A ; X, +X 2 E B} .JX~EAX1 +X2EBThis establishes (4) .To prove (5), we begin by observing that the second equation has just beenproved . Next we observe that since {X 1 , . . . , X,) and {S1, . . . , S, } obviouslygenerate the same Borel field, the left member of (5) is justJq'{Sn = B I X1, . . . , Xn-1 } .Now it is trivial that as a function of (XI, . . ., Xi_ 1 ), S, "depends on themonly through their sum S,_," . It thus appears obvious that the first term in (5)should depend only on Sn _ 1 , namely belong to ~ JS,,_,) (rather than the larger:'f {S1, . . . , S i _ 1 }) . Hence the equality of the first and third terms in (5) shouldbe a consequence of the assertion (13) in Theorem 9 .1 .5 . This argument,however, is not rigorous, and requires the following formal substantiation .Let µ(") = Al x . . .X An = A(n-1) X An andn-1A = n S 71 (Bj ),1= 1