UA MATH563 The Central Limit Theorem of the Mathematical Foundation of Probability Theory 22 The Portmanteau Theorem of Weak Convergence in Metric Probability Space

Now we discuss weak convergence in the metric space, assuming $(Oh,d)$ is a metric space,$(Oh,\mathcal{F},P)$ is a probability space,$X_n,X$ is defined inΩ \OmegaRandom variables on$\Omega$ , their distribution is${\mu }_n,\mu$。

Portmanteau's theorem
With regard to convergence in distribution, the following statements are equivalent:

1. $X_n{\to }_{d}X$
2. For any open set $G$，$lim\mathrm{i}\mathrm{n}\mathrm{f}P(X_n\in G)\ge P(X\in G)$
3. For any closed set $K$，$lim\mathrm{s}\mathrm{u}\mathrm{p}P(X_n\in K)\le P(X\in K)$
4. AA for any set$A$ , if$P(X\in \partial A)=0$，则$\lim P(X_n\in A)=P(X\in A)$

Regarding weak convergence, the following statements are equivalent:

1. ${\mu }_n\Rightarrow \mu$
2. For any open set $G$，$lim\mathrm{i}\mathrm{n}\mathrm{f}{\mu }_n(G)\ge \mu \left(G\right)$
3. For any closed set $K$，$lim\mathrm{s}\mathrm{u}\mathrm{p}{\mu }_n(K)\le \mu \left(K\right)$
4. AA for any set$A$ , if$\mu (\partial A)=0$，则$\mu (A_n)\to \mu \left(A\right)$

The path of proof is $1\Rightarrow 3\Rightarrow 2\Rightarrow 4\Rightarrow 1.$ Post a Durrett certificate