Two independent weak convergence sequences have dependent limit? Let $X_n\Rightarrow X$ and $Y_n\Rightarrow Y$ as $n\to\infty$, where all these $X$ and $Y$ are well defined random variables and $\Rightarrow$ is the convergence in distribution.
Could you give me an example in which $X_n$ and $Y_n$ are independent, but $X$ and $Y$ are not independent?
Thank you!

Let $a$ be such that $F_X(x)$ is continuous at $a$, and $b$ be such that $F_Y(x)$ is continuous at $b$. Then $F_{X_n}(a)\to F_X(a)$, $F_{Y_n}(b)\to F_Y(b)$. On the other hand,
$$
\mathbb{P}(X_n\le a,Y_n\le b)=\mathbb{P}(X_n\le a)\ \mathbb{P}(Y_n\le b)
$$
by independence. As a result, the LHS of the above
$$
F_{X_n,Y_n}(a,b)\to F_X(a) F_Y(b)
$$
so if $F_{X,Y}(a,b):=F_X(a)F_Y(b)$ then
$$
\mathbb{P}(X_n\le a,Y_n\le b)\to F_{X,Y}(a,b)\quad\text{as }n\to \infty
$$
and thus the pair $(X_n,Y_n)$ converges in distribution to a random variable whose CDF is given by $F_{X,Y}(a,b)$, and given its product form, $X$ and $Y$ are independent.
The only thing which remains to consider is the case when $F_X(x)$ is not continuous at $a$ (and similar for $Y$).
