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The classic result is for iid sequences x i i 12 with

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The classic result is for i.i.d. sequences X i : i 1,2,. .. with E X i 2 . The CLT does not hold if E X i 2 .Let E X i and 2 Var X i . As before, let X ̄ n n 1 i 1 n X i be the sample average of the first n variables. By the WLLN, X ̄ n p . The CLT is about the standardized sample average. 78
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THEOREM :Let X i : i 1,2,. .. be i.i.d. with E X i 2 , and let E X i , 2 Var X i . Then W n X ̄ n / n n X ̄ n d Normal 0,1 . We will not prove this but use it as a tool. We can also write the standardized variable W n in terms of the sum Y n : W n Y n n n Y n E Y n  SD Y n d Normal 79
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Remarkably, this result holds for any distribution with a finite second moment, including discrete distributions. Suppose the X i are i.i.d. Bernoulli p RVs. Then Y n X 1 ... X n ~ Binomial n , p .Wewe can compute probabilities such as P Y n y using the binomial density. The CLT says that we can approximate this probability: P Y n y P Y n np y np P Y n np np 1 p y np np 1 p y np np 1 p for “large” n by the CLT. 80
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If instead the X i are i.i.d. Poisson random variables, the CLT approximation is P Y n y y n n We can compare this with exact probabilities using Y n Poisson n . We sometimes abuse the notion of convergence in distribution and write X ̄ n a ~ Normal , 2 / n but this is only to be viewed as a shorthand for the formal statement of the CLT. 81
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We need not standardize by to get a limiting normal distribution. It is immediate that n X ̄ n d Normal 0, 2 The CLT implies that X ̄ n O p n 1/2 when 0 because then n X ̄ n converges in distribution, and so n X ̄ n O p 1 . We showed this via Chebyshev’s inequality earlier. 82
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Multivariate CLT Now let X i : i 1,2,. .. be a sequence of i.i.d. k 1 random vectors with E X ij 2 , j 1,. .., k .Let E X i and let Var X i ; assume is positive definite. Then the vector of sample averages, X ̄ n , has E X ̄ n Var X ̄ n Var n 1 i 1 n X i n 2 i 1 n Var X i n 2 i 1 n / n . As in the univariate case, Var X ̄ n converges to zero at rate 1/ n .
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The classic result is for iid sequences X i i 12 with E X i...

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