依分布收敛、依概率收敛、均方收敛、几乎处处收敛

1. 分布函数列的弱收敛性

弱收敛 {Fn(x)}\{F_n(x)\}{Fn(x)} 是分布函数列，如果存在一个函数 F(x)F_(x)F(x), 对任意 xxx , 使得 lim⁡n→∞Fn(x)=F(x)\lim_{n \to \infty}F_n(x)=F(x)limn→∞Fn(x)=F(x), 则称Fn(x)F_n(x)Fn(x) 弱收敛于 F(x)F(x)F(x), 并记为
Fn(x)→wF(x).F_n(x)~\overset{w}{\underset{}{\to}}~F(x).Fn(x) →w F(x).

2. 随机变量的四种收敛性

假定如下的随机变量 XXX 与随机变量序列 {Xn}\{X_n\}{Xn} 定义在概率空间 (Ω,F,P)(\Omega,F,P)(Ω,F,P) 上.

依分布收敛 设 XnX_nXn 的分布函数为 Fn(x)F_n(x)Fn(x), XXX 的分布函数为 F(x)F(x)F(x), 假如
Fn(x)→wF(x)，F_n(x)\overset{w}{\underset{}{\to}}F(x)，Fn(x)→wF(x)，
则称{Xn}\{X_n\}{Xn} 依分布收敛于 XXX, 记为Xn→dX.X_n~\overset{d}{\underset{}{\to}}~X.Xn →d X.

依分布收敛为这里讨论收敛性中最弱的收敛，其只能保证分布函数序列收敛。

依概率收敛 若对 ∀\forall∀ ε>0\varepsilon>0ε>0 ,
lim⁡n→∞P{w:∣Xn(w)−X(w)∣≥ε}=0,\lim_{n \to \infty}P\{w: |X_n(w)-X(w)| \geq \varepsilon\}=0,n→∞limP{w:∣Xn(w)−X(w)∣≥ε}=0,
则称{Xn}\{X_n\}{Xn} 依概率收敛于 XXX, 记为Xn→PX.X_n~\overset{P}{\underset{}{\to}}~X.Xn →P X.

依概率收敛强于依分布收敛，其先给定ε>0\varepsilon>0ε>0，保证 ∣Xn(w)−X(w)∣≥ε|X_n(w)-X(w)| \geq \varepsilon∣Xn(w)−X(w)∣≥ε 随着 nnn 增大，概率趋向于0。这里可见 ε\varepsilonε 越小收敛性越好。

r-阶收敛 假设 E∣X∣r<∞E|X|^r<\inftyE∣X∣r<∞, E∣Xn∣r<∞,r>0E|X_n|^r<\infty, r>0E∣Xn∣r<∞,r>0. 如果
lim⁡n→∞E∣Xn−X∣r=0，\lim_{n \to \infty}E|X_n-X|^r=0，n→∞limE∣Xn−X∣r=0，
则称{Xn}\{X_n\}{Xn} r-阶收敛于 XXX。其中当r=2r=2r=2时，称为均方收敛。

r-阶收敛强于依概率收敛，其说明随着 nnn 的增大，r-阶的 ∣Xn−X∣|X_n-X|∣Xn−X∣ 的期望是为0的。

几乎处处收敛（依概率1收敛） 如果
P(w:lim⁡n→∞Xn(w)=X(w))=1，P(w:\lim_{n \to \infty}X_n(w)=X(w))=1，P(w:n→∞limXn(w)=X(w))=1，
称{Xn}\{X_n\}{Xn} 几乎处处收敛于 XXX, 记为 Xn→a.s.X.X_n~\overset{a.s.}{\underset{}{\to}}~X.Xn →a.s. X.

几乎处处收敛也强于依概率收敛，和依概率收敛比较，可以看成取定ε=0\varepsilon=0ε=0，所以也叫依概率1收敛。事实上，几乎处处收敛是不收敛点的集合是只构成勒贝格零集。

3. 收敛性间关系

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推出

几乎处处收敛

依概率收敛

均方收敛

依分布收敛

参考资料

[1] 刘嘉焜，王公恕，应用随机过程（第二版）。