Convergence in distribution on product space (definition)

Question

Let $S$ and $T$ be seperable metric spaces, $(X_n)_{n\in\mathbb{N}}$ a $S$-valued sequence of random variables and $(Y_n)_{n\in\mathbb{N}}$ a $T$-valued sequence of random variables

Is convergence in distribution on a product space defined by

$\mathbf{E}[f(X_n)g(Y_n)]\to\mathbf{E}[f(X)g(Y)]$

for all continuous, bounded functions $f:S\to\mathbb{R}$, $g:T\to\mathbb{R}$

?

Answer 1

No, we should consider $S \times T$ with corresponding metric and check the next condition: $\mathbf{E}[f(X_n, Y_n)]\to\mathbf{E}[f(X, Y)]$.

Here we suppose that $(X_n, Y_n)$ are defined on one probability space $\Omega_n$. We have similar condition for $(X,Y)$.