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01 Convergence of Random Variabl

01 Convergence of Random Variabl

作者: 顾劝劝 | 来源:发表于2021-06-29 17:08 被阅读0次

本章提要:

  • 各种收敛定义
  • continuous mapping 和 Slutsky's theorems
  • big-O
  • 主要的收敛理论

收敛定义

依概率收敛

1 狭义距离

Let X_n be a sequence of random vectors. Then X_n converges in probability to X,
X_n\xrightarrow{p} X
if for all \epsilon>0,
\mathbb{P} (||X_n-X||\geq\epsilon)\rightarrow 0 \text{ as } n \rightarrow \infty

2 广义度量

我们把||\cdot|| 换成\rho(\cdot),上述定义变成
\mathbb{P} (\rho(X_n-X)\geq\epsilon)\rightarrow 0 \text{ as } n \rightarrow \infty

依分布收敛(也叫Weak convergence)

1

For random variables X_n\in\mathbb R and X\in \mathbb R, X_n converges in distribution to X,
X_n\xrightarrow{d} X \text{ or } X_n \leadsto X
if for all x such that x\mapsto \mathbb P(X \leq x) is continuous,
\mathbb P(X_n\leq x)\rightarrow \mathbb P(X\leq x) \text{ as } n\rightarrow \infty

2 对于度量空间-valued r.v.

For metric space-valued random variables X_n,\ X,\ X_n converges in distribution to X if for all bounded continuous f
\mathbb E[f(X_n)]\rightarrow \mathbb E[f(X)] \text{ as } n \rightarrow \infty

3 L^p 收敛(p阶均值收敛)

X_n \xrightarrow{L^p} X if \lim_{n\rightarrow \infty} \mathbb E[||X_n-X||^p]\rightarrow 0

几乎绝对收敛

Random variables converge almost surely, X_n \xrightarrow{a.s.} X, if
\mathbb P(\lim_n X_n \neq X)=0 \text{ or } \mathbb P(\lim_n ||X_n-X||\geq \epsilon)=0

它们之间的关系

X_n\xrightarrow{a.s.}X \Rightarrow X_n\xrightarrow{p}X \Rightarrow X_n\xrightarrow{d}X
X_n\xrightarrow{L^p}X\Rightarrow X_n\xrightarrow{p}X

众所周知的强大数定律(均值几乎绝对收敛到期望)和中心极限定律(中心化的均值乘\sqrt n依分布收敛到方差为原生成机制协方差矩阵的零均值正态分布)就不再赘述了。

Portmanteau Lemma(依分布收敛形式还有n胞胎)

以下皆等价:
(1) X_n\xrightarrow{d}X
(2) \mathbb E[f(X_n)]\rightarrow \mathbb E[f(X)] for all bounded continuous f
(3) \mathbb E[f(X_n)]\rightarrow \mathbb E[f(X)] for all 1-Lipschitz f with f(x)\in[0,1]
(4) \lim \inf_n \mathbb E[f(X_n)]\geq \mathbb E[f(X)] for all continuous nonnegative f
(5) ...

Continuous mapping theorems

Let g be continuous on a set B such that \mathbb P(X\in B)=1. Then
(1) X_n\xrightarrow{p}X implies g(X_n)\xrightarrow{p}g(X)
(2) X_n\xrightarrow{d}X implies g(X_n)\xrightarrow{d}g(X)
(3) X_n\xrightarrow{a.s.}X implies g(X_n)\xrightarrow{a.s.}g(X)

Slutsky's theorems

注意,前面的X是随机变量,这里的c是常数
(1)X_n\xrightarrow{d}c iff X_n\xrightarrow{p} c
(2)X_n\xrightarrow{d}X and d(X_n,Y_n)\xrightarrow{p}0 implies Y_n\xrightarrow{p} X
(3) X_n\xrightarrow{d}X and Y_n\xrightarrow{p} c implies
\binom{X_n}{Y_n}\xrightarrow{d}\binom{X}{c}

那我们怎么用它?引理在此
If X_n\xrightarrow{d}X and Y_n\xrightarrow{d}c, then
(1) X_n+Y_n\xrightarrow{d} X+c
(2) X_nY_n\xrightarrow{d}cX
(3) If C\in\mathbb R^{d\times d} with det(C)\neq 0, and Y_n\xrightarrow{d}C, then Y_n^{-1}X_n \xrightarrow{d} C^{-1}X

根据这个Slutsky定理我们可以知道t分布T_n \xrightarrow{d} N(0,1)

大O

定义

R_n是随机变量
X_n=Y_nR_n, Y_n\xrightarrow{p}0 \Rightarrow X_n=o_P(R_n)
X_n=Y_nR_n, Y_n\xrightarrow{p}O_P(1) \Rightarrow X_n=O_P(R_n)
Y_n=O_P(1)意思是Y_n uniformly tight, 也就是
\lim_{M\rightarrow \infty} \sup_n \mathbb P(|||Y_n||\geq M)=0

引理

小op1相加还是小op1,大Op1相加还是大Op1,小opX=大OpX

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