Theorem 192 let xn be a sequence of random variables a

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Unformatted text preview: ose that (Ω, F , P) is a probability space, and let X1 , X2 , . . ., X , and Y all be real-valued random variables on (Ω, F , P). (a) L1 is a vector space and expectation is a linear map on L1 . Furthermore, expectation is positive. That is, if X , Y ∈ L1 with 0 ≤ X ≤ Y , then 0 ≤ E(X ) ≤ E(Y ). 19–1 (b) X ∈ L1 if and only if |X | L1 , in which case we have |E(X )| ≤ E(|X |). (c) If X = Y almost surely (i.e., if P {ω : X (ω ) = Y (ω )} = 1), then E(X ) = E(Y ). (d) (Monotone Convergence Theorem) If the random variables Xn ≥ 0 for all n and Xn ↑ X (i.e., Xn → X and Xn ≤ Xn+1 ), then ￿ ￿ lim E(Xn ) = E lim Xn = E(X ). n→∞ n→∞ (We allow E(X ) = +∞ if necessary.) (e) (Fatou’s Lemma) If the random variables Xn all satisfy Xn ≥ Y almost surely for some Y ∈ L1 and for all n, then ￿ ￿ E lim inf Xn ≤ lim inf E(Xn ). (19.1) n→∞ n→∞ In particular, (19.1) holds if Xn ≥ 0 for all n. (f ) (Lebesgue’s Dominated Convergence Theorem) If the random variables Xn → X , and if for some Y ∈ L1 we have |Xn | ≤ Y almost...
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