目錄

• • 1. 分布函數列的弱收斂性
  • 2. 隨機變量的四種收斂性
  • 3. 收斂性間關系
  • 參考資料

1. 分布函數列的弱收斂性

弱收斂 $\{F_n(x)\}$ 是分布函數列，如果存在一個函數 $F_{(}x)$, 對任意 $x$ , 使得 ${\lim }_{n\to \infty }{F}_n(x)=F(x)$, 則稱$F_n(x)$ 弱收斂于 $F(x)$, 并記為
$$F_n(x)\stackrel{w}{\to }F(x).$$

2. 隨機變量的四種收斂性

假定如下的隨機變量 $X$ 與隨機變量序列 $\{X_n\}$ 定義在概率空間 $(\Omega ,F,P)$ 上.

依分布收斂 設 $X_n$ 的分布函數為 $F_n(x)$, $X$ 的分布函數為 $F(x)$, 假如
$$F_n(x)\stackrel{w}{\to }F(x)，$$
則稱$\{X_n\}$ 依分布收斂于 $X$, 記為$X_n\stackrel{d}{\to }X.$

• 依分布收斂為這里討論收斂性中最弱的收斂，其只能保證分布函數序列收斂。

依概率收斂 若對 $?$ $\epsilon >0$ ,
$$\underset{n\to \infty }{\lim }P\{w:|X_n(w)?X(w)|\ge \epsilon \}=0,$$
則稱$\{X_n\}$ 依概率收斂于 $X$, 記為$X_n\stackrel{P}{\to }X.$

• 依概率收斂強于依分布收斂，其先給定$\epsilon >0$，保證 $|X_n(w)?X(w)|\ge \epsilon$ 隨著 $n$ 增大，概率趨向于0。這里可見 $\epsilon$ 越小收斂性越好。

r-階收斂 假設 $E|X{|}^r<\infty$, $E|X_n{|}^r<\infty ,r>0$. 如果
$$\underset{n\to \infty }{\lim }E|X_n?X{|}^r=0，$$
則稱$\{X_n\}$ r-階收斂于 $X$。 其中當$r=2$時，稱為均方收斂。

• r-階收斂強于依概率收斂，其說明隨著 $n$ 的增大，r-階的 $|X_n?X|$ 的期望是為0的。

幾乎處處收斂（依概率1收斂） 如果
$$P(w:\underset{n\to \infty }{\lim }{X}_n(w)=X(w))=1，$$
稱$\{X_n\}$ 幾乎處處收斂于 $X$, 記為 $$X_n\stackrel{a.s.}{\to }X.$$

• 幾乎處處收斂也強于依概率收斂，和依概率收斂比較，可以看成取定$\epsilon =0$，所以也叫依概率1收斂。事實上，幾乎處處收斂是不收斂點的集合是只構成勒貝格零集。

3. 收斂性間關系

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參考資料

[1] 劉嘉焜，王公恕，應用隨機過程（第二版）。

總結

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