I understand how to create random variables with a prespecified correlational structure using a Cholsesky decomposition. But I would like to be able to solve the inverse problem: Given random variables $X_1, X_3, \dots X_n$ ,and two different linear sums of those variables $V_1=a_{11}X_1+a_{12}X_2+\dots a_{1n}X_n$, and $V_2=a_{21}X_1 + a_{22}X_2 +\dots+a_{2n}X_n$, I wish to calculate the correlation between the $V_1$ and $V_2$.
I have searched for terms like "linear combination random variables correlation" and have found plenty of material discussing how the correlation affects the variance of the sum of random variables. Unfortunately I have found nothing that seems to relate to the problem described. I would appreciate any information at all, including either an appropriate book chapter or web page reference.
Best Answer
The covariance is bilinear, hence $$ \mathrm{Corr}(V_1,V_2)=\frac{\mathrm{Cov}(V_1,V_2)}{\sqrt{\mathrm{Var}(V_1)\mathrm{Var}(V_2)}}=\frac{\sum\limits_{j=1}^n\sum\limits_{k=1}^na_{1j}a_{2k}\mathrm{Cov}(X_j,X_k)}{\sqrt{\mathrm{Var}(V_1)\mathrm{Var}(V_2)}}, $$ where, for $i=1$ and $i=2$, $$ \mathrm{Var}(V_i)=\sum\limits_{j=1}^n\sum\limits_{k=1}^na_{ij}a_{ik}\mathrm{Cov}(X_j,X_k). $$