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.
Best Answer
Let $$Y_n:=\sum_{\substack{i,j=1\\ i\neq j}}^nX_iX_j.$$ We can rewrite it as $$\left(\sum_{k=1}^nX_k\right)^2-\sum_{k=1}^nX_k^2.$$ This suggests the use of well known limit theorems.