I have to calculate the following expectation
$$\mathbb{E}\left[\left(\frac1M\sum\limits_{i=1}^MX(i-n_1-M)\right)\left(\frac1M\sum\limits_{j=1}^MX(j-n_2-M)\right)\right]$$
where $M$, $n_1$ and $n_2$ are constant, known values, and $X(n)$ is a random white, Gaussian process that has zero mean and unitary variance.
So far what I did was
$$\frac{1}{M^2}\sum\limits_{i=1}^M\sum\limits_{j=1}^M\mathbb{E}\left[X(j-n_2-M)X(i-n_1-M)\right]=\frac{1}{M^2}\sum\limits_{i=1}^M\sum\limits_{j=1}^M\delta(i-j+n_1-n_2)$$
where $\delta(n)$ is the Dirac delta function, i.e.
$$\delta (n) = \left\{ \begin{array}{ll} 1 & \mbox{if } n= 0 \\ 0 & \mbox{if } n \neq 0 \end{array} \right.$$
I don't know how to solve that double summation to get something shorter, if possible. Any ideas?