By Gian-Carlo Rota, Kenneth Baclawski

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**Extra resources for An Introduction to Probability and Random Processes**

**Example text**

Proof: Since (i) ~ (2) by Theorem fices to note that the implication as in (2). with o n Define by induction C S, o > n n -- la and (2) => O n sense sequence ib, it suf- In fact, of stopping times let S be (an)nf N such that E~fO[X ] -~ [I the strict (i) by Theorem (3) is always valid. an increasin$ o,T C S } and let Sn = S(On). (3) ~ < -- ~ < -- 1 - xo Ill -< 7 T Then the sequence (Sn)nf N is decreasing, Sn is abundant (see Lemma 2 in §2) and the set {X T ] T C S n} is Ll-bounded. in 60 Let now o,T be arbitrary elements longs to % and on this set T = o V Then oV T C Sn, the set {o < T} be- in Sn.

We assume throughout that 8 is infinite-dimensional. Conditions for Absolute Continuity Determined by Sequences A general method for determining if absolute continuity holds is based on the following well-known result. Lemma i. Let {Fn, n ~ I} be an increasing family of sub-u-fields of BIB] such that B[S] is the smallest G-field containing nzlU Fn. , ~) to F . , ~n ) be the restric- Then v << p if and only if (a) ~n << ~n for all n ~ 1 [b) {d~n/d~n, n ~ i> is uniformly integrable with respect to ~.

Also Therefore f(xT - X°)dP = Af(X°V T - X = f Definition Then A T )dP + fAc (Xo AT - X o V T )dP [Xov ~] - XO A T )dP + f (XoA T- E gOAT Ac [Xov ~] )dP, whence IIf ( x This proves - X)dPII _< IIE'~A~ [X v ~] - X A • I{1" the first part of the lemma. Assume now that E = R. In this case it is clear sup f(X T - Xo)dP = (~,~) If(xT - Xo)dP ]. (o,T) c~,z¢S a,~¢S Hence to prove sup that II) we need only show that Ef O[X T] - X~l]1 __< sup 1[ (o,~) a < • S sup (o,~) f ( x - Xo)dP. o,T CS ~,~ To see this, let O,T C S, o _< T and let B = {X O _< E °[XT]}.