Can statistical dependence arise only at the limit?

Question

(This has been inspired by a comment exchange with @guy).

Assume we have two infinite sequences of random variables, $\{X_n\}$ and $\{Y_n\}$. Assume the RVs in $\{X_n\}$ are statistically independent from the RVs in $\{Y_n\}$. Assume that $X_n \to_d X$ and $Y_n \to_d Y$.

Is it possible that $X$ and $Y$ are dependent?

If yes, an example?

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@guy noted that convergence in distribution is a statement about what is the distribution of the limiting random variable, not what is the limiting random variable. So it would appear that nothing is implied about dependence/independence...
Well, intuitively, while "gradual loss of dependence:" and hence "asymptotic independence" is easy to imagine, I cannot see how dependence is absent for all finite values of the index and then suddenly it emerges at the limit... maybe cases involving discontinuities could do the trick.

Answer 1

I think I see the disconnect between our positions. The following statement is true.

Let $X_n$ and $Y_n$ be sequences of independent random variables such that $X_n \to X^\star$ and $Y_n \to Y^\star$ in distribution. Then there exists random variables $X$ and $Y$, with $X$ independent of $Y$, such that $(X_n, Y_n) \to (X, Y)$ in distribution such that $X$ and $Y$ marginally have the same distributions as $X^\star$ and $Y^\star$.

The point I am trying to make is that the following statement is not true.

Let $X_n$ and $Y_n$ be sequences of independent random variables, and let $X$ and $Y$ be random variables such that $X_n \to X$ and $Y_n \to Y$ in distribution. Then $X$ and $Y$ are independent.

The reason is that there is no constraint imposed on $(X,Y)$ in the second statement, whereas the first statement explicitly postulates $X$ and $Y$ independent.

Answer 2

We assume independence for all finite $n$, so

$$P(X_n \leq x,Y_n\leq y) = P(X_n \leq x)\cdot P(Y_n\leq y)$$

or

$$F_n(x,y) = G_n(x)\cdot H_n(y)$$

We want to determine

$$\lim_{n \to \infty} F_n(x,y) = \lim_{n \to \infty} \big[G_n(x)\cdot H_n(y)\big]$$

under the additional assumptions that

$$G_n(x) \to G(x),\;\;\; H_n(x) \to H(x)$$

So to preserve independence at the limit, we want the conditions under which

$$\lim_{n \to \infty} \big[G_n(x)\cdot H_n(y)\big] = G(x)H(x)$$

which is a familiar "convergence of product of function sequences" case.

Then, by standard results, to ensure the result we need at least one of the two marginal distribution function sequences to converge uniformly, and not just pointwise (so that we can say that the product converges pointwise). (we also need both of them to be bounded, and distribution functions are).

Moreover, we also know that, if the limiting distribution function has no points of discontinuity, then the convergence is uniform (see this post).

It appears then that we can state:

Let two independent sequences of random variables converge in distribution. If one of the two limiting distribution functions is continuous, then independence is preserved at the limit.

It follows that in cases where both limiting distribution functions have discontinuities, it appears that dependence can in principle arise at the limit.