Central Limit Theorem

对任意的$y_{t}\sim IID(0,\sigma^{2})$, 令其样本数为T, 则有: $\sqrt{T}\cdot \bar y \xrightarrow{d}N(0,\sigma^{2})$, for $\overline{y}=\frac{1}{T}\sum_{t=1}^{T}y_{t}$.

CLT 有许多版本, 但这个版本是最火的.

Law of Large Numbers

对任意的$y_{t}\sim IID(\mu,\sigma^{2})$, 令其样本数为T, 则有: $\overline{y}\xrightarrow{d}E[y_{t}]=\mu$. 即sample mean converges to population mean.

Slutsky Theorem

任意的变量$a_{t}\xrightarrow{d}A$, for A is distribution or random variable, 任意的变量$b_{t}\xrightarrow{p}c,for\ c\ is\ constant$.

则有: $\frac{a_{t}}{b_{t}}\xrightarrow{d}\frac{A}{c}$.