Suppose I have a random variable: $$ Y_{it}=A_{it}+B_{it} $$ which is made up of two random variables $A$ and $B.$ $i$ is the unit of observation that runs through $i=1....N$ for each time period $t=1...T$. Now taking the covariance of $Y$ for any two units $i$ and $j$ over time, we have: $$ cov(Y_{i},Y_{j})=cov(A_{i},A_{j})+cov(B_{i},B_{j})+cov(A_{i},B_{j})+cov(A_{j},B_{i}) $$ Now, I want to figure out the average contribution of the term $cov(A_{i},A_{j})$ to the term $cov(Y_{i},Y_{J}).$ What I have is $$ C=\frac{1}{N^2}\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{cov(A_{i},A_{j})}{cov(Y_{i},Y_{j})} $$
Notice that the variance-covariance matrix is symmetric. As such, there are ionly $N/(N-1)/2$ unique entries ($cov(A_{1},A_{_2}$)=$cov(A_{2},A_{1})$. I know that the expression above is incorrect (as it generates duplicates). How can I modify the above expression to correct notation such that it contains only a unique occurence of symmetric covariances?