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

Throughout, the laws $P_{X_{n}}$ and $P_{X}$ are Borel probability measures on $S$ (The Law of a Random Element (i)).

Step 1 (expectations as integrals). Let $f\in C_{b}(S)$. Then $f$ is Borel measurable and integrable against every finite Borel measure on $S$ (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)), so Expectations Computed from the Law (ii), applied on $(\Omega_{n},\mathcal{F}_{n},P_{n})$ and on $(\Omega,\mathcal{F},P)$, gives

$$\mathbb{E}_{n}f(X_{n})=\int_{S}f\,dP_{X_{n}},\qquad\mathbb{E}f(X)=\int_{S}f\,dP_{X}.$$

Step 2 (proof of (i)). By Convergence in Distribution of Random Elements of a Metric Space (i), $X_{n}\xrightarrow{d}X$ means $P_{X_{n}}\Rightarrow P_{X}$, that is, $\int_{S}f\,dP_{X_{n}}\to\int_{S}f\,dP_{X}$ for every $f\in C_{b}(S)$ (Weak Convergence of Finite Borel Measures on a Metric Space (i)); by Step 1 this gives the condition of (i). Conversely, assume that condition. Every bounded Lipschitz $f:S\to\mathbb{R}$ lies in $C_{b}(S)$ (Bounded Continuous Functions on a Metric Space: Measurability, Integrals, Operations, Lipschitz Functions and Approximation of Closed Sets (vi)), so the condition contains the condition of (ii), which by Step 1 says that $\int_{S}f\,dP_{X_{n}}\to\int_{S}f\,dP_{X}$ for every bounded Lipschitz $f:S\to\mathbb{R}$. The portmanteau theorem, from Portmanteau Theorem on a Metric Space (ii) to Portmanteau Theorem on a Metric Space (i), gives $P_{X_{n}}\Rightarrow P_{X}$, that is, $X_{n}\xrightarrow{d}X$.

Step 3 (proof of (ii)). If $X_{n}\xrightarrow{d}X$, then by Step 2 $\mathbb{E}_{n}f(X_{n})\to\mathbb{E}f(X)$ for every $f\in C_{b}(S)$, in particular for every bounded Lipschitz $f$, as these lie in $C_{b}(S)$. Conversely, assume the condition of (ii); by Step 1 it says $\int_{S}f\,dP_{X_{n}}\to\int_{S}f\,dP_{X}$ for every bounded Lipschitz $f:S\to\mathbb{R}$. The laws form a sequence $(P_{X_{n}})$ of functions $\mathcal{B}(S)\to[0,\infty]$, each a Borel probability measure, so the portmanteau theorem, the implication from Portmanteau Theorem on a Metric Space (ii) to Portmanteau Theorem on a Metric Space (i), gives $P_{X_{n}}\Rightarrow P_{X}$, that is, $X_{n}\xrightarrow{d}X$ by Convergence in Distribution of Random Elements of a Metric Space (i).