Suppose $\{Z_t\}$ is i.i.d. $(\mu,\sigma^2)$, and $ \bar{Z}_n=n^{-1}\sum_{t=1}^n Z_t$, then as $n\rightarrow \infty$,
\[\frac{\bar{Z}_n-E(\bar{Z}_n)}{\sqrt{var(\bar{Z}_n)}}=\frac{\bar{Z}_n-\mu}{\sqrt{\sigma^2/n}}\]
\[=\frac{\sqrt{n}(\bar{Z}_n-\mu)}{\sigma}\]
\[\rightarrow^d N(0,1)\]
證明：設 $Y_t=\frac{Z_t-\mu}{\sigma}$ 和 $\bar{Y}_n=n^{-1}\sum_{t=1}^nY_t$. 則有
\[\frac{\sqrt{n}(\bar{Z}_n-\mu)}{\sigma}=\sqrt{n}\bar{Y}_n.\]
$\sqrt{n}\bar{Y}_n$ 的特征函數為：
\[\phi_n(u)=E(\exp(iu\sqrt{n}\bar{Y}_n)),\quad i=\sqrt{-1}\]
\[=E(exp(\frac{iu}{\sqrt{n}}\sum_{t=1}^nY_t))\]
\[=\Pi_{t=1}^nE(exp(\frac{iu}{\sqrt{n}}Yt))\quad \text{by independence}\]
\[=(\phi_Y(\frac{u}{\sqrt{n}}))^n \quad \text{by identical distribution}.\]
\[=(\phi_Y(0)+\phi'(0)\frac{u}{\sqrt{n}}+\frac{1}{2}\phi''(0)\frac{u^2}{n}+\cdots)^n\]
\[=(1-\frac{u^2}{2n})^n+o(1)\]
\[\rightarrow \exp(-\frac{u^2}{2})\quad as\quad n\rightarrow \infty\]
由特征函數與分布函數一一對應關系和 $\exp(-\frac{u^2}{2})$ 是標准正態分布的特征函數，即得證。