N n combining these two inequalities gives ex eu

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Unformatted text preview: simple with Yn,k ↑ Xn and E(Yn,k ) ↑ E(Xn ) as k → ∞ which is possible by Propositions 18.1 and 18.2. Set Zk = max Yn,k , n ≤k and observe that 0 ≤ Zk ≤ Zk+1 so that (Zk ) is an increasing sequence of non-negative simple random variables which necessarily has a limit Z = lim Zk . k→∞ By Proposition 18.2, E(Zk ) ↑ E(Z ). We now observe that if n ≤ k , then Yn,k ≤ Zk ≤ Xn ≤ Xk ≤ X. (23.1) We now deduce from (23.1) that Xn ≤ Z ≤ X almost surely, and so letting n → ∞ implies X = Z almost surely. We also deduce from (23.1) that E(Yn,k ) ≤ E(Zk ) ≤ E(Xk ) for n ≤ k . Fix n and let k → ∞ to obtain E(Xn ) ≤ E(Z ) ≤ lim E(Xk ). k→∞ Now let n → ∞ to obtain lim E(Xn ) ≤ E(Z ) ≤ lim E(Xk ). n→∞ k→∞ Thus, E(Z ) = lim E(Xn ). n→∞ But X = Z almost surely so that E(X ) = E(Z ) and we conclude E(X ) = lim E(Xn ) n→∞ as required. 23–1 Theorem 23.2 (Fatou’s Lemma). Suppose that (Ω, F , P) is a probability space, and let X1 , X2 , . . . be real-value...
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This document was uploaded on 04/07/2014.

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