If xn are general random variables with k1 exk then

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Unformatted text preview: d random variables on (Ω, F , P). If Xn ≥ Y almost surely for some Y ∈ L1 and for all n, then ￿ ￿ E lim inf Xn ≤ lim inf E(Xn ). (23.2) n→∞ n→∞ In particular, (23.2) holds if Xn ≥ 0 for all n. ˜ Proof. Without loss of generality, we can assume Xn ≥ 0. Here is the reason. If Xn = Xn −Y , ˜ n ∈ L1 with Xn ≥ 0 and ˜ then X ￿ ￿ ￿ ￿ ˜ ˜ E lim inf Xn ≤ lim inf E(Xn ) if and only if E lim inf Xn ≤ lim inf E(Xn ) n→∞ n→∞ because n→∞ ￿ ￿ ˜ lim inf Xn = lim inf Xn − Y. n→∞ Hence, if Xn ≥ 0, set n→∞ n→∞ Yn = inf Xk k ≥n so that Yn ≥ 0 is a random variable and Yn ≤ Yn+1 . This implies that Yn converges and lim Yn = lim inf Xn . n→∞ n→∞ Since Xn ≥ Yn , monotonicity of expectation implies E(Xn ) ≥ E(Yn ). This yields ￿ ￿ lim inf E(Xn ) ≤ lim E(Yn ) = E lim Yn = lim inf E(Xn ) n→∞ n→∞ n→∞ n→∞ by the Monotone Convergence Theorem and the proof is complete. Theorem 23.3 (Lebesgue’s Dominate...
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This document was uploaded on 04/07/2014.

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