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Unformatted text preview: Lecture 8: Weak Law of Large Numbers 8-1 Lecture 8 : Weak Law of Large Numbers References: Durrett [Sections 1.4, 1.5] The Weak Law of Large Numbers is a statement about sums of independent ran- dom variables. Before we state the WLLN, it is necessary to define convergence in probability. We say Y n converges in probability to Y and write Y n P-→ Y if, ∀ ² > 0, P ( ω : | Y n ( ω )- Y ( ω ) | > ² ) → , n → ∞ . Theorem 8.4.3 (Weak Law of Large Numbers) Let X,X 1 ,X 2 ,... be a sequence of i.i.d. random variables with E | X | 2 < ∞ and define S n = X 1 + X 2 + ··· + X n . Then S n n P-→ E X. Proof: In this proof, we employ the common strategy of first proving the result under an L 2 condition (i.e. assuming that the second moment is finite), and then using truncation to get rid of the extraneous moment condition. First, we assume E ( X 2 ) < ∞ . Because the X i are iid, Var S n n ¶ = 1 n 2 n X i =1 Var( X i ) = Var( X ) n ....
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This note was uploaded on 11/18/2009 for the course MATH 241 taught by Professor Reed during the Spring '09 term at Duke.
- Spring '09
- Law Of Large Numbers