中心极限定理证明（独立同分布）

Suppose $\{Z_t\}$ is i.i.d. $(\mu,\sigma^2)$, and $ \bar{Z}_n=n^{-1}\sum_{t=1}^n Z_t$, then as $n\rightarrow \infty$,

\[\frac{\bar{Z}_n-E(\bar{Z}_n)}{\sqrt{var(\bar{Z}_n)}}=\frac{\bar{Z}_n-\mu}{\sqrt{\sigma^2/n}}\]

\[=\frac{\sqrt{n}(\bar{Z}_n-\mu)}{\sigma}\]

\[\rightarrow^d N(0,1)\]

证明：设 $Y_t=\frac{Z_t-\mu}{\sigma}$ 和 $\bar{Y}_n=n^{-1}\sum_{t=1}^nY_t$. 则有

\[\frac{\sqrt{n}(\bar{Z}_n-\mu)}{\sigma}=\sqrt{n}\bar{Y}_n.\]

$\sqrt{n}\bar{Y}_n$ 的特征函数为：

\[\phi_n(u)=E(\exp(iu\sqrt{n}\bar{Y}_n)),\quad i=\sqrt{-1}\]

\[=E(exp(\frac{iu}{\sqrt{n}}\sum_{t=1}^nY_t))\]

\[=\Pi_{t=1}^nE(exp(\frac{iu}{\sqrt{n}}Yt))\quad \text{by independence}\]

\[=(\phi_Y(\frac{u}{\sqrt{n}}))^n \quad \text{by identical distribution}.\]

\[=(\phi_Y(0)+\phi'(0)\frac{u}{\sqrt{n}}+\frac{1}{2}\phi''(0)\frac{u^2}{n}+\cdots)^n\]

\[=(1-\frac{u^2}{2n})^n+o(1)\]

\[\rightarrow \exp(-\frac{u^2}{2})\quad as\quad n\rightarrow \infty\]

由特征函数与分布函数一一对应关系和 $\exp(-\frac{u^2}{2})$ 是标准正态分布的特征函数，即得证。