Question about summation notation index I'm working through a proof involving the sum of covariances but the notation is tripping me up.  What does it mean when you are taking a summation over the index $i < j$?  For instance $\sum_{i < j}\mathrm{Cov}(X_{i},X_{j})$ where Cov is covariance. 
I'm sure it's nothing complicated I just want to make sure I understand. 
 A: The notation means that you sum over all allowed $i$ and $j$ such that $i < j$. The notation is compact and protects you from errors that may come from doing the end-points incorrectly.
Here is an example. Suppose that $1\leq i \leq m$ and $1\leq j \leq n$ where $m$ and $n$ are not the same. If $m < n$, the sum becomes
$$\sum_{i < j} a_{ij} = \sum_{j = i+1}^n \sum_{i=1}^{m}a_{ij}.$$
If $n < m$ we have
$$\sum_{i < j} a_{ij} = \sum_{j = i+1}^n \sum_{i=1}^{n-1}a_{ij}.$$
Formally, these are two different expressions and to determine which one you are using can take a few seconds here but it could take a lot more time for more complicated sums.
A: An Illustration
Suppose you have $i\in I=\{1,2,\cdots,5\}$ and $j\in J=\{1,2,\cdots,6\}$. The following picture shows you which of the terms $\sum\limits_{i<j} a_{i,j}$ include.

According to the picture, if you choose a certain value of $i$, then $j$ should at least be $i+1$, otherwise the condition $i<j$ isn't met (falling on or below the line $i=j$).
In conclusion,
$$\sum\limits_{i<j} a_{i,j} = \sum_{i} \sum_{\text{All $j$'s that's $>i$}} a_{i,j}.$$
If you sum $j$ last, you get
$$\sum\limits_{i<j} a_{i,j} = \sum_{j} \sum_{\text{All $i$'s that's $<j$}} a_{i,j}.$$
