Let \(X_{n}, X, Y: \Omega ightarrow \mathbb{R}, n \geqslant 1\), be random variables. If

\[\lim _{n ightarrow \infty} \mathbb{E}\left(f\left(X_{n}ight) g(Y)ight)=\mathbb{E}(f(X) g(Y)) \quad \text { for all } f \in \mathcal{C}_{b}(\mathbb{R}), g \in \mathcal{B}_{b}(\mathbb{R}) \text {, }\]

then \(\left(X_{n}, Yight) \xrightarrow{d}(X, Y)\). If \(X=\phi(Y)\) for some \(\phi \in \mathcal{B}(\mathbb{R})\), then \(X_{n} \xrightarrow{\mathbb{P}} X\).