Can anybody tell me why it holds that:
I got it from my teacher, who left for vacation. I'm really confused by the change in indices.
Look at some simple examples, with, say, T= 3. The outer sum then goes t= 1, t= 2, t= 3: for each such t, the term being added is
(Adding "1" from y= 1 to t just gives "t")
That entire sum is .
On the right side, again with T= 3, y takes on values of 1, 2, 3 and the inner sum there is
Multiplying each of those by "1" and adding gives , exactly the same.
For any T, both right and left hand sides give .
The summation is over the integer points in the triangular region in the picture. If you start by fixing y, then you sum over t (along a horizontal line) with t going from y to T (and then you take the results and sum them over y going from 1 to T). But if you start by fixing t, then you sum over y (along a vertical line) with y going from 1 to t (and then you take the results and sum them over t going from 1 to T).
Edit. This is just another way of saying what HallsofIvy has explained, but using a graphical picture rather than an algebraic illustration.