Theorem (Portmanteau for Probability Measures)

Let $\mu_{n},\mu$ be Probability Measures on $(\Omega,\mathcal{F})$. The following conditions are equivalent:

1. $\mu_{n}$ converges weakly to $\mu$, i.e. for every Continuous and bounded $f:\Omega\to \mathbb{R}$:$$\int\limits _{\mathbb{X}}f(x) \, \mu_{n}(dx)\to \int\limits _{\mathbb{X}}f(x) \, \mu(dx) .$$
2. For all $x\in \mathbb{R}$ s.t. $\mu \{ x \}=0$ $$\mu_{n}((-\infty,x])\to\mu((-\infty,x])$$
3. For every Closed set $C\in\Omega$: $$\limsup_{ n \to \infty }\mu_{n}(C)\le \mu(C) .$$
4. For every Open set $G$: $$\liminf_{ n \to \infty }\mu_{n}(G)\ge \mu(G).$$
5. For every $B\in\mathcal{B}(\Omega)$ whose boundary has $\mu$-Measure $0$ (i.e. $\mu(\partial B)=0$):$$\lim_{ n \to \infty } \mu_{n}(B)=\mu(B).$$

Theorem (Portmanteau for Random Variables)

For rvs $X,X_{1},X_{2},\dots$ on $(\Omega,\mathcal{F},\mathbb{P})$, the following are equivalent:

1. $X_{n}$ converges to $X$ in distribution, i.e. for all $t\in \mathbb{R}$ such that $F(t)$ is Continuous $$F_{n}(t)\to F(t)$$

2. For all Continuous and bounded $f:\Omega\to \mathbb{R}$ $$\mathbb{E}[f(X_{n})]\to \mathbb{E}[f(X)]$$

3. For every Closed set $F\subseteq \mathbb{R}$ $$\mathbb{P}(X\in F)\ge \limsup_{ n \to \infty } \mathbb{P}(X_{n}\in F)$$

4. For every Open set $G\subseteq \mathbb{R}$ $$\mathbb{P}(X\in G)\le \liminf_{ n \to \infty } \mathbb{P}(X_{n}\in G)$$

5. For every $B\in\mathcal{B}$ such that $\mathbb{P}(X\in \partial B)=0$ $$\mathbb{P}(X_{n}\in B)\to \mathbb{P}(X\in B)$$

6. For any $\psi \in C_{c}^{2}(\mathbb{R})$ $$\mathbb{E}[\psi(X_{n})]\to \mathbb{E}[\psi(X)]$$