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83 the multivariate clt implies that var x n 12 x n

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The multivariate CLT implies that Var X ̄ n  1/2 X ̄ n 1/2 n X ̄ n d Normal 0 , I k Note that the standardized random vector Z n 1/2 n X ̄ n has E Z n 0 and Var Z n I k . We can instead write n X ̄ n d Normal 0 , 84
8 . Some Additional Useful Limit Results Equipped with the multivariate CLT and the algebra of convergence in probability and convergence in distribution, we can state several limit results that have application for asymptotic analysis. (i) Suppose A n is a seqence of random r k matrices with A n p A , and W n is a sequence of k 1 random vectors with W n d W . Then A n W n d AW 85

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To show this, write A n W n AW n A n A W n AW n o p 1 O p 1 AW n o p 1 and use AW n d AW along with the asymptotic equivalence lemma. If W ~ Normal 0 , then A n W n d Normal 0 , A A This usually applies when W n n 1/2 X ̄ n and the CLT is in force, and then A n n 1/2 X ̄ n  d Normal 0 , A A 86
(ii) Under the same setting above with W Normal 0 , , nonsingular, and rank A r , W n A n A A 1 A n W n d r 2 This follows because the quadratic function is continuous in A n W n and W A A A 1 AW r 2 . (iii) In the same setup as part (ii), let n be a sequence of symmetric, PSD matrices with n p . Then W n A n A n n A n 1 A n W n d r 2 87

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This is a little harder to show, but we use the asymptotic equivalance lemma by showing that W n A n A n n A n 1 A n W n W n A n A A 1 A n W n o p 1 . To see this, note that W n A n A n n A n 1 A n W n W n A n A A 1 A n W n W n A n  A n n A n 1 A A 1 A n W n O p 1  o p 1 O p 1 because A n n A n 1 A A 1 p 0 . Now we can apply the asymptotic equivalence lemma. 88
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83 The multivariate CLT implies that Var X n 12 X n 12 n X...

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