1 i 1 n x i n 2 i 1 n var x i n 2 i 1 n n as in the

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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 . 83
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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
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