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Unformatted text preview: d Convergence Theorem). Suppose that (Ω, F , P) is
a probability space, and let X1 , X2 , . . . be real-valued random variables on (Ω, F , P). If the
random variables Xn → X , and if for some Y ∈ L1 we have |Xn | ≤ Y almost surely for all
n, then Xn ∈ L1 , X ∈ L1 , and
lim E(Xn ) = E(X ).
n→∞ Proof. Suppose that we deﬁne the random variables U and V by
U = lim inf Xn and V = lim sup Xn .
n→∞ n→∞ The assumption that Xn → X implies that U = V = X almost surely so that E(U ) =
E(V ) = E(X ). And if we also assume that |Xn | ≤ Y almost surely for all n so that Xn ∈ L1 ,
then we conclude |X | ≤ Y . Since Y ∈ L1 , we have X ∈ L1 . Moreover, Xn ≥ −Y almost
surely and −Y ∈ L1 so by Fatou’s Lemma,
E(U ) ≤ lim inf E(Xn ).
n→∞ 23–2 However, we also know −Xn ≥ −Y almost surely and
−V = lim inf (−Xn )
n→∞ so Fatou’s lemma also implies
−E(V ) = E(−V ) ≥ lim inf E(−Xn ) = − lim sup E(Xn ).
n→∞ n→∞ Combining the...
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