<< statistics

Limit Theorem

1. Central Limit Theorem

We say the random variable list $\{X_n,n\ge 1\}$ obys the central limit theorem, if the $S_n$ approaches the normal distribution when $n\to \mathrm{\infty }$. That is, the cumulative distribution function $F_n(x)$ of distribution the standardized $S_n$ meets $${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{\displaystyle \frac{S_n-\text{E}{S}_n}{\sqrt{\text{Var}{S}_n}}}=\mathrm{\Phi }(x)}$$ where $\mathrm{\Phi }(x)$ is the cumulative distribution function of the standard normal distribution.

For the IID $\{X_n,n\ge 1\}$,

2. Law of Large Number

Suppose the random variable $X,X_1,X_2,...$ defined at the same probability space $(\mathrm{\Omega },\mathcal{F},P)$, if we have $$\underset{n\to \mathrm{\infty }}{lim}P(|X_n-X|\ge ϵ)=0$$ for any $ϵ>0$, we say the random variable list $\{X_n,n\ge 1\}$ converges in probability, denoted as $X_n\stackrel{P}{\to }X$.

We say the random variable list $\{X_n,n\ge 1\}$ obeys the law of large numbers, if $$\frac{S_n-\text{E}{S}_n}{n}\stackrel{P}{\to }0$$ where ${\displaystyle S_n=\sum _{k=1}^n{X}_k}$.