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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 ).
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
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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