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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 \(n\rightarrow\infty\). That is, the cumulative distribution function \(F_n(x)\) of distribution the standardized \(S_n\) meets \[ \displaystyle \lim_{n\rightarrow\infty} \dfrac{S_n - \E S_n}{\sqrt{\Var S_n}} = \Phi(x) \] where \(\Phi(x)\) is the cumulative distribution function of the standard normal distribution.
For the IID \(\{X_n, n\ge 1\}\),
Suppose the random variable \(X, X_1, X_2, ...\) defined at the same probability space \((\Omega, \mathcal{F}, P)\), if we have \[ \lim_{n\rightarrow\infty} P(|X_n-X|\ge \epsilon) = 0 \] for any \(\epsilon > 0\), we say the random variable list \(\{X_n, n\ge1\}\) converges in probability, denoted as \(X_n \xrightarrow[]{P} X\).
We say the random variable list \(\{X_n, n\ge 1\}\) obeys the law of large numbers, if \[ \frac{S_n - \E S_n}{n}\xrightarrow[]{P} 0 \] where \(\displaystyle S_n = \sum_{k=1}^{n} X_k\).