Two sequences $X_1, X_2, \ldots, Y_1, Y_2,\ldots : (\Omega, \mathcal{F},\mathbb{P}) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ of real random variables such that $\forall n \ X_n, Y_n $ are independent are given. Show that if $X_n \to X$, $Y_n \to Y$ then $X, Y$ are independent, too.
I'd be grateful for any help.
Two words about this convergence: $X_n \to X \iff \mathbb{P}(\{\omega \in \Omega : X_n(\omega) \to X(\omega)\})=1.$
Hint By Kac's theorem, two random variables $X$ and $Y$ are independent if, and only if, $$\mathbb{E}\exp(\imath \xi X+\imath \eta Y) = \mathbb{E}\exp(\imath \, \xi X) \cdot \mathbb{E}\exp(\imath \, \eta Y) \tag{1}$$ for any $\xi,\eta \in \mathbb{R}$. Use the dominated convergence theorem to prove $(1)$ under the given assumptions.
I believe one can prove this using characteristic functions.
I would use two facts:
$ \ $
1. Lévy-Cramer's Theorem Link to Wikipedia
Simplifying just a little, it states that:
$ Z_n \rightarrow Z \iff \forall t \in \mathbb{R} \ \ \phi_{Z_n}(t) \rightarrow \phi_{Z}(t) \ \ \ (pointwise) $
where $Z$'s are random variables and $\phi$'s are corresponding characteristic functions.
2. Second one is that if $X$ and $Y$ are random variables then:
$ X, Y $ are independent $ \iff \forall (s,t) \in \mathbb{R}^2 \ \ \phi_{(X,Y)}(s,t) = \phi_{X}(s) \phi_{Y}(t) $
$ \ $
Now, with these two facts known, for any $s, t \in \mathbb{R}$ we can write:
$ \phi_{(X,Y)}(s,t) = \lim \limits_{n \to \infty} \phi_{(X_n,Y_n)}(s,t) = \lim \limits_{n \to \infty} \phi_{X_n}(s) \phi_{Y_n}(t) = \lim \limits_{n \to \infty} \phi_{X_n}(s) \lim \limits_{n \to \infty} \phi_{Y_n}(t) = \phi_{X}(s) \phi_{Y}(t) $
Second equality follows from the fact that $X_n$ and $Y_n$ are independent (here we use second fact). First and last equality follows from the first fact - Lévy-Cramer Theorem.
