7.8 Suppose X1, X2,... and Y1, Y2,... are two sequences of random variables, each sequence i.i.d., and suppose that the random variables X1 and Y1 are correlated with correlation coefficient ρ, while Xi and Yj are independent if i = j. With the usual notations X¯ n, Y¯n, denote Sn(X) = 81 n

n i=1(Xi − X¯ n)2 and Sn(Y) =

81 n

n i=1(Yi − Y¯n)2. The sample correlation coefficient is defined as rn =

1 n

n i=1(Xi − X¯ n)(Yi − Y¯n)

Sn(X)Sn(Y) =

1 n

n i=1 XiYi − X¯ nY¯n Sn(X)Sn(Y) .

Show that there exists some ν constant such that

√n(rn − ρ) D

−→ N(0, ν2).

Show that in the particular case when the pairs (Xi, Yi) are i.i.d. multivariate normal,



Xi Yi



∼ MVN2

μX

μY



,



σ2 X ρσXσY

ρσXσY σ2 Y



The calculation of ν is explicit and

√n(rn − ρ) D

−→ N(0,(1 − ρ2)

2).