Consider a sequence of i.i.d. random variables, $(X_i)_{i=1}^n$, with zero mean and unit variance.
I want to calculate the limit (a.s.) of
$$ \frac{1}{n}\sum_{i\neq j}X_iX_j $$ as $n\to\infty$.
My initial guess was that this sum converge to $0$. But it can be seen that the variance is given by $$ \text{var}\left(\frac{1}{n}\sum_{i\neq j}X_iX_j \right) = \frac{n-1}{n}\mathbb{E}\left(X_1^2X_2^2\right) $$ which don't goes to zero, and so this guess is wrong.