When does marginal tightness imply joint tightness? Suppose a sequence of random variables $\{X_n\}$ converges in distribution to some probability measure $\mu_X$ on $\mathbb{R}$, and similarly $Y_n \stackrel{d}{\Rightarrow} \mu_Y$.
When is it true that $(X_n,Y_n)$ also converges in distribution? Or, more generally, when is $(X_n,Y_n)$ tight?
 A: Tightness of marginals implies joint tightness in the product topology. This is readily seen from the definition of tightness, recalling that the product of two compacts is compact.
A: If $(\mu_n)_{n \in \mathbb{N}}$ is a sequence of Borel probability measures on $X \times X$ endowed with the product topology for a Polish space $X$, then compactness of the corresponding marginal sequences $(\mu_n^{(1)})_n$ and $(\mu_n^{(2)})_n$ implies tightness of  $(\mu_n)_{n\geq 1}$ as follows:
Let $\epsilon > 0$ and $K_i \subseteq X$ compact such that $\mu_n^{(i)}(K_i) \geq 1-\frac{\epsilon}{2}$ (such sets exist by the assumed tightness of both marginal sequences). Then, $K_1 \times K_2 \subseteq X\times X$ is compact and for each $n \geq 1$:
$$\mu_n((K_1\times K_2)^c) \leq \mu_n(X\times K_2^c)+\mu_n(K_1^c \times X)=\mu_n^{(2)}(K_2^c)+\mu_n^{(1)}(K_1^c) \leq \epsilon.$$Hence, $(\mu_n)_n$ is tight.
This applies to your question by considering $\mu_n^{(1)} \sim X_n$, $\mu_n^{(2)} \sim Y_n$ and $\mu_n \sim (X_n,Y_n)$, noting that by your assumption both these marginal sequences are tight.
