E 2 n 0 53 limits and integrals this section presents

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E 2 n 0 5.3 Limits and Integrals This section presents a sequence of results which describe how expectation and limits interact. Under certain circumstances we are allowed to interchange expec- tation and limits. In this section we will learn when this is safe. Theorem 5.3.1 (Monotone Convergence Theorem (MCT)) If 0 Xn t X then 0 E(Xn) t E(X). This was proved in the previous subsection 5.2.3. See 3 page 123. Corollary 5.3.1 (Series Version ofMCT) If 0 are non-negative random variables for n 1, then 00 00 = j=l j=l so that the expectation and infinite sum can be interchanged. To see this, just write n = lim t £(" n_.oo j=l (MCT) n = lim t n_.oo j=l 00 = j=l 0
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132 5. Integration and Expectation Theorem 5.3.2 (Fatou Lemma) If X n 0, then £(lim infXn) ::::: lim inf E(Xn). n-+oo n-+00 More generally, if there exists Z e Lt and Xn Z, then E(liminfXn)::::: liminfE(Xn) . n-+oo n-+oo Proof of Fatou. If X n 0, then E (lim infXn) = E t (6 x,)) n-+00 = t E (3xk) (from MCI' 5.3.1) ::::: lim inf E(Xn). n-+oo For the case where we assume X n Z, we have X n - Z 0 and E (liminf(Xn- Z))::::: liminfE(Xn- Z) n-+00 n-+oo so E(liminfXn)- E(Z)::::: liminfE(Xn)- E(Z). n-+oo n-+oo The result follows by cancelling E(Z) from both sides of the last relation. 0 Corollary 5.3.2 (More Fatou) If Xn ::::: Z where Z e Lt, then E(limsupXn) limsupE(Xn). n-+oo n-+oo Proof. This follows quickly from the previous Fatou Lemma 5.3.2. If Xn ::::: Z, then -Xn -Z e L., and the Fatou Lemma 5.3.2 gives E(liminf(-Xn))::::: liminf£(-Xn), n-+00 n-+oo so that £(-liminf(-Xn)) -liminf(-EXn) . n-+oo n-+oo The proof is completed by using the relation - lim inf- = lim sup . 0 Canonical Example. This example is typical of what can go wrong when limits and integrals are interchanged without any dominating condition. Usually some- thing very nasty happens on a small set and the degree of nastiness overpowers the degree of smallness.
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5.3 Limits and Integrals 133 Let (Q, B, P) = ([0, 1], 8((0, 1]), .l.) where, as usual, ). is Lebesgue measure . Define Xn = n 2 1(0.1/n)· For any we (0 , 1] , so However so and 1(0, 1/n)(w) --+ 0, Xn--+ 0. 1 E(Xn) = n 2 · - = n--+ oo, n E(liminfXn) = 0 < liminf(EXn) = oo n-+00 n-+oo E(limsupXn) = 0, limsupE(Xn) = 00. n-+00 n-+oo So the second part of the Fatou Lemma given in Corollary 5.3 .2 fails. So obviously we cannot hope for Corollary 5. 3.2 to hold without any restriction. 0 Theorem 5.3.3 (Dominated Convergence Theorem (DCT)) If Xn--+ X, and there exists a dominating random variable Z E L 1 such that IXnl then E(Xn)--+ E(X) and EIXn- XI--+ 0. Proof of DCT. This is an easy consequence of the Fatou Lemma. We have -Z ::5 Xn ::5 Z and - Z e L 1 as well as Z e L So both parts of Fatou 's lemma apply: E(X) =E(liminfXn) n-+oo :::;lim inf E(Xn) n-+oo :::;limsupE(Xn) n-+oo supXn) n-+oo =E(X). (Fatou Lemma 5.3.2) (since inf < sup) (Corollary 5.3. 2) Thus all inequalities are equality. The rest follows from IX n - X I ::::; 2Z . o
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134 5. Integration and Expectation 5.4 Indefinite Integrals Indefinite integrals allow for integration over only part of the Q-space. They are simply defined in terms of indicator functions.
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