Solved – Correlation of linear combination of random variables

The variables are a random sample – i.e. they are independent. Covariance of independent variables is zero, so you can eliminate $Cov(X_i, X_j)$ where $ineq j$.

From the property of covariance, you obtain: $Cov(U_m, U_n)=sum_{i=1}^m sum_{j=1}^n a_i a_j Cov(X_i, X_j)= sum_{i=1}^m a_i^2 Cov(X_i, X_i)=sum_{i=1}^m a_i^2 sigma^2$ (because $m leq n$).

For correlation, lets calculate $Var(S_n), Var(S_m)$. Variance of sum of i.i.d. R.V.s equals to sum of their variances, hence $Var(S_n)=sum_{i=1}^n a_i^2 sigma^2$, and similarly for $S_m$. Then:

$rho(S_n, S_m)=frac{sum_{i=1}^m a_i^2 sigma^2 }{sqrt{sum_{i=1}^n a_i^2 sigma^2}sqrt{sum_{i=1}^m a_i^2 sigma^2}}$. It can be seen that if $m=n$, then correlation is 1.

If $a_i=frac{1}{n}$, then $Cov(U_m, U_n)=frac{m}{n^2}sigma^2$ and correlation is $frac{m}{sqrt{mn}}$.