Do copulas preserve uniform convergence?

Question

Suppose I have a distribution $Q$ of $\left(Y_i, X_i\right)$, such that $Q(y,x)=C(F_Y(y), F_X(x))$, for some copula $C$ and marginal CDFs $F_X$ and $F_Y$, by Sklar's theorem.

Now, suppose I have sequences of CDFs $F^{(y)}_n$ and $F^{(x)}_n$ such that $\Vert F^{(y)}_n - F_Y \Vert_\infty \rightarrow 0$, $\Vert F^{(x)}_n - F_X \Vert_\infty \rightarrow 0$.

Now, we can define $Q_n = C(F^{(y)}_n, F^{(x)}_n)$. My question is, do we have $\Vert Q_n - Q\Vert_\infty \rightarrow 0$?

Answer 1

Concerning the limit of copulas, I think you may have forgotten the fact that every copula is Lipschitz wrt to $l^1$-norm, with Lipschitz coefficient is $1$. That is, for any $u,v \in [0,1]^N$, we have: $$ |C(u)-C(v)| \le \sum_{k=1}^N |u_k-v_k|$$

Your desired limit is just a direct consequence.