Convergence in Distribution through Expectations of Bounded Continuous and of Bounded Lipschitz Functions

In the setting of Convergence of Probability Measures and Random Elements: Standing Notation, for each $n\in\mathbb{N}$ let $X_{n}$ be a random element of $S$ on a probability space $(\Omega_{n},\mathcal{F}_{n},P_{n})$, with $\mathbb{E}_{n}$ the expectation there, and let $X$ be a random element of $S$ on $(\Omega,\mathcal{F},P)$. For $f\in C_{b}(S)$ the random variables $f(X_{n})$ and $f(X)$ are integrable (Bounded Continuous Functions on a Metric Space: Measurability, Integrals, Operations, Lipschitz Functions and Approximation of Closed Sets (i), Bounded Continuous Functions on a Metric Space: Measurability, Integrals, Operations, Lipschitz Functions and Approximation of Closed Sets (ii), Expectations Computed from the Law (ii)).

(i) Bounded continuous functions. $X_{n}\xrightarrow{d}X$ if and only if

$$\mathbb{E}_{n}f(X_{n})\to\mathbb{E}f(X)\qquad(n\to\infty)$$

for every $f\in C_{b}(S)$.

(ii) Bounded Lipschitz functions. $X_{n}\xrightarrow{d}X$ if and only if $\mathbb{E}_{n}f(X_{n})\to\mathbb{E}f(X)$ for every bounded Lipschitz function $f:S\to\mathbb{R}$, which lies in $C_{b}(S)$ by Bounded Continuous Functions on a Metric Space: Measurability, Integrals, Operations, Lipschitz Functions and Approximation of Closed Sets (vi).