# strong law of large number with Covariance

I'm supposed to prove that $$\frac{1}{n} \sum_{j=1}^n (X_j-\overline X_n)(Y_j-\overline Y_n)$$ converges almost surely to $$Cov(X,Y)$$ assuming that $$(X_i,Y_i)$$ are iid with the same distribution as $$(X,Y)$$ for $$i=1,\dots,n$$ . Clearly that is the same as proving that $$\frac{1}{n} \sum_{j=1}^n X_j Y_j$$ converges almost surely to $$E[XY]$$ because $$\frac{1}{n} \sum_{j=1}^n (X_j-\overline X_n)(Y_j-\overline Y_n) = \frac{1}{n} \sum_{j=1}^n X_j Y_j - \overline X_n \overline Y_n$$ as $$\overline X_n \overline Y_n$$ converges almost surely to $$E[X]E[Y]$$ and we have the formula $$Cov(X,Y)=E[XY]-E[X]E[Y]$$
However I'm not sure how to prove that because it'll require to prove that the $$X_1Y_1,\dots,X_nY_n$$ are i.i.d. (if we use the strong law of large number) and I don't know how to prove that...any hints ? Thans you

Note : $$\overline X_n = \frac{1}{n}\sum_{j=1}^n X_j$$ (same for $$\overline{Y_n}$$)

• Forgot to mention: X,Y have finite positive variances (not sure how I can use that) Commented Mar 3, 2021 at 14:32

For any $$t\in\mathbb R$$, let $$D_t=\{(x,y): xy \leqslant t\}$$. This is Borel set in $$\mathbb R^2$$. Then $$F_{X_1Y_1}(t)=\mathbb P(X_1Y_1 \leqslant t) = \mathbb P((X_1,Y_1) \in D_t)=\mathbb P((X_2,Y_2) \in D_t)=\mathbb P(X_2Y_2 \leqslant t)=F_{X_2Y_2}(t).$$ The central equality is valid since $$(X_1,Y_1)$$ has the same distribution as $$(X_2,Y_2)$$.
So we obtain that $$X_1Y_1,\ldots, X_nY_n$$ are identically distributed.
Next, for any $$t,s$$ $$\mathbb P(X_1Y_1 \leqslant t,\, X_2Y_2 \leqslant s) = \mathbb P((X_1,Y_1) \in D_t,\, (X_2,Y_2) \in D_s)$$ (use independence of $$(X_1,Y_1)$$ and $$(X_2,Y_2)$$) $$= \mathbb P((X_1,Y_1) \in D_t)\cdot \mathbb P( (X_2,Y_2) \in D_s) = \mathbb P(X_1Y_1 \leqslant t) \cdot \mathbb P(X_2Y_2 \leqslant s).$$
So we get independence of $$X_1Y_1, X_2Y_2$$. The same way independence of $$X_1Y_1,\ldots, X_nY_n$$ can be proved.