Limit law of real-valued independent random variables

Question

Let $X_n$ and $Y_n$ be real-valued independent r.vs, each of whose limit law is $X$ and Y, resp.

i.e $X_n \overset{d}{\to} X$ and $Y_n \overset{d}{\to} Y$ for some r.vs $X$ and $Y$.

Then, are $X$ and $Y$ independent as well?

I don't think it holds, but with the lemma below, I make it, which makes me surprised.

Lemma. $X_{1}, \cdots, X_{n}$ are independent if and only if $$ \Bbb{E}[ f_{1}(X_{1})\cdots f_{n}(X_{n}) ] = \Bbb{E} f_{1}(X_{1}) \cdots \Bbb{E} f_{n}(X_{n})$$ for any bounded continuous functions $f_{1}, \cdots, f_{n}$.

Thus, I wonder if there exists such result.

Anyone, any comments would be helpful. Thanks in advance.

Answer 1

Then, are $X$ and $Y$ independent as well?

Of course not, consider some nondegenerate random variable $X$, independent sequences $(X_n)$ and $(Y_n)$ i.i.d. distributed like $X$, and $Y=X$.

How you planned to apply the lemma is a mystery.

Answer 2

I should leave this as a comment but I don't have enough points. I assume that $\forall (m,n) \in \mathbb{N}^2$, $X_n \perp Y_m$.

You can use the the properties of the characteristic function of the couple $(X_n, Y_m)$ and Levy's continuity theorem. (I suggest working with the characteristic function because it is continuous, so it's neat not to worry about treating discontinuity points).