strong law of large number with Covariance

Question

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 $\overline{Y_n}$)

Answer 1

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.