Question: Problem 6. Suppose X1, X2, are i.i.d random variables taking values in R 2 with Sn = X1 + + Xn. Assume 2x1 matrix EX1

Problem 6. Suppose X1, X2, are i.i.d random variables taking values in R 2 with Sn = X1 + + Xn.

Assume 2x1 matrix EX1 = (0) 1x1 element is 0 and 2x1 element is 1

(1)

2x2 matrix Cov(X) = ( 1 0 )

( 0 1 )

1x1 =1, 1x2=0, 2x1=0 , 2x2=1

(i) Derive the CLT for Sn/ n. Derive the CLT for (Sn/sqrt(n)

(ii) Prove (||Sn||^2) /n --> 2 (2) by Problem 5 this is chi square with degrees of freedom =2

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