851_lectures17_24

# Note that the proofs generally follow the so called

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Unformatted text preview: surely for all n, then Xn ∈ L1 , X ∈ L1 , and lim E(Xn ) = E(X ). n→∞ Remark. This theorem contains ALL of the central results of Lebesgue integration theory. Theorem 19.2. Let Xn be a sequence of random variables. (a) If Xn ≥ 0 for all n, then E ￿ ∞ ￿ n=1 Xn ￿ = ∞ ￿ E( Xn ) n=1 with both sides simultaneously being either ﬁnite or inﬁnite. (b) If ∞ ￿ n=1 then E(|Xn |) < ∞, ∞ ￿ Xn n=1 converges almost surely to some random variable Y ∈ L1 . In other words, ∞ ￿ n=1 19–2 Xn (19.2) is integrable with E ￿ ∞ ￿ Xn n=1 ￿ = ∞ ￿ E(Xn ) . n=1 Thus, (19.2) holds with both sides being ﬁnite. Notation. For 1 ≤ p < ∞, let Lp = {random variables X : Ω → R such that |X |p ∈ L1 }. Theorem 19.3 (Cauchy-Schwartz Inequality). If X , Y ∈ L2 , then XY ∈ L1 and ￿ |E(XY )| ≤ E(X 2 )E(Y 2 ). Theorem 19.4. Let X : (Ω, F , P) → (R, B ) be a random variable. (a) (Markov’s Inequality) If X ∈ L1 , then P {|X | ≥ a} ≤ E(|X |) a for every a > 0. (b) (Chebychev’...
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## This document was uploaded on 04/07/2014.

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