# Weak convergence in metric space

We give some definitions and basic properties of weak convergence of probability measures on metric spaces, based on [Bil99].

Let ${(S, \mathcal{B}(S))}$ be a metric space. ${\mathbb{P}_n, \mathbb{P}}$ are probability mesures on ${(S, \mathbb{B}(S))}$. We write ${\mathbb{P} f = \int_S f d\mathbb{P}}$ for ${\mathcal{B}(S)/\mathcal{B}(\mathbb{R})}$ measurable ${f}$ for convenience.

Definition 1 ${\mathbb{P}_n}$ is said to be weakly convergent to ${\mathbb{P}}$, denoted by ${\mathbb{P}_n \Rightarrow \mathbb{P}}$, if ${\mathbb{P}_n f \rightarrow \mathbb{P} f}$ for every ${f\in C_b(S; \mathbb{R})}$.

The following theorem provides useful conditions equivalent to definition of weak convergence. A set ${A\in \mathbb{B}(S)}$ is called ${\mathbb{P}}$-continuity set, if ${\mathbb{P}(\partial A) = 0}$.

Theorem 2 (The Portmanteau Theorem, [Bil99, Theorem 2.1]) These five conditions are equivalent:

1. ${\mathbb{P}_n \Rightarrow \mathbb{P}}$.
2. ${\mathbb{P}_n f \rightarrow \mathbb{P} f}$ for all bounded uniformly continuous ${f}$.
3. ${\lim\sup_n \mathbb{P}_n F \le \mathbb{P} F}$ for all closed ${F}$
4. ${\lim\inf_n \mathbb{P}_n G \ge \mathbb{P} G}$ for all open ${G}$.
5. ${\mathbb{P}_n A \rightarrow \mathbb{P} A}$ for all ${\mathbb{P}}$-continuity sets ${A}$.

Let ${\mathcal{P}}$ be the space of probability measures on ${\mathcal{B}(S)}$. ${\mathbb{P}}$ can be metrized by Prohov distance ${\pi(P, Q)}$ ([Bil99, P. 72]) if ${S}$ is separable and complete metric space.

Theorem 3 ([Bil99, Theorem 6.8]) Suppose that ${S}$ is separable and complete. Then

1. weak convergence is equivalent to ${\pi}$-convergence,
2. ${\mathcal{P}}$ is separable and complete
3. ${\mathcal{P}}$ is relatively compact if and only if its ${\pi}$-closure is ${\pi}$-compact.

Let ${(S', \mathcal{B}(S'))}$ be another metric space, and ${h:S \rightarrow S'}$ is measurable ${\mathcal{B}(S)/\mathcal{B}(S')}$. Then ${\mathbb{P}h^{-1}}$ is a probability measure on ${\mathcal{B}(S')}$.

Theorem 4 (Mapping theorem, [Bil99, Theorem 2.7]) Let ${h:S \rightarrow S'}$ is measurable ${\mathcal{B}(S)/\mathcal{B}(S')}$, and ${\mathbb{P}_n \Rightarrow \mathbb{P}}$. In addition, ${\mathbb{P} D_h = 0}$, where ${D_h \in \mathcal{B}(S)}$ be the set of discontinuities of ${h}$. Then ${\mathbb{P}_n h^{-1} \Rightarrow \mathbb{P} h^{-1}}$.

The above mapping theorem saids, weak convergence of probability measures are preserved by mapping ${h}$, if ${h}$ is continuous almost surely with respect to the limit measure ${\mathbb{P}}$, i.e. ${\mathbb{P}(D_h) = 0}$.

Let ${(\Omega, \mathcal{F}, \mathbb{P})}$ be a probability space, and ${X:\Omega \rightarrow S}$ be measurable w.r.t. ${\mathcal{F}/\mathcal{B}(S)}$. Then, ${X}$ is called random element. The distribution of ${X}$ is the probability measure ${\mathbb{P}X^{-1}}$ on ${(\Omega, \mathcal{F})}$, which is sometimes denoted by ${\mathcal{L}(X)}$. If ${f: S \rightarrow \mathbb{R}}$ is a measurable ${\mathcal{B}(S)/ \mathcal{B}(\mathbb{R}))}$, then ${\mathbb{E} [f(X)] = \mathbb{P}X^{-1}f}$.

Definition 5 We say a sequence ${\{X_n\}}$ converges in distribution to ${X}$, denoted by ${X_n \Rightarrow X}$, if ${\mathbb{P}X_n^{-1} \Rightarrow \mathbb{P}X^{-1}}$.

Thus, ${X_n \Rightarrow X}$ if and only if ${\mathbb{E}[f(X_n)] \rightarrow \mathbb{E}[f(X)]}$ for all ${f\in C_b(S)}$. We call a set ${A \in \mathcal{B}(S)}$ as an ${X}$-continuity set if ${\mathbb{P}(X\in \partial A) = 0}$. Now, we can rewrite Theorem 2 in terms of random elements.

Theorem 6 ([Bil99, p. 26]) These five conditions are equivalent:

1. ${X_n \Rightarrow X}$.
2. ${\mathbb{E}[ f(X_n)] \rightarrow \mathbb{E} [f(X)]}$ for all bounded uniformly continuous ${f}$.
3. ${\lim\sup_n \mathbb{P}(X_n \in F) \le \mathbb{P} (X\in F)}$ for all closed ${F}$
4. ${\lim\inf_n \mathbb{P} (X_n \in G) \ge \mathbb{P} (X\in G)}$ for all open ${G}$.
5. ${\mathbb{P} (X_n\in A) \rightarrow \mathbb{P} (X\in A)}$ for all ${X}$-continuity sets ${A}$.

Also, the Theorem 4 can be rewritten into mapping theorem of random elements: ${X_n \Rightarrow X}$ on ${S}$ implies ${h(X_n) \Rightarrow h(X)}$ on ${S'}$, if ${h}$ is continuous almost surely with respect to ${X}$, i.e. ${\mathbb{P}(X\in D_h) = 0.}$