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 be a sequence of random vectors. Then converges in probability to ,
if for all ,
2 广义度量
我们把 换成,上述定义变成
依分布收敛(也叫Weak convergence)
1
For random variables and , converges in distribution to ,
if for all such that is continuous,
2 对于度量空间-valued r.v.
For metric space-valued random variables converges in distribution to if for all bounded continuous
3 收敛(p阶均值收敛)
if
几乎绝对收敛
Random variables converge almost surely, , if
它们之间的关系
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)
(2) for all bounded continuous
(3) for all 1-Lipschitz with
(4) for all continuous nonnegative
(5) ...
Continuous mapping theorems
Let be continuous on a set such that . Then
(1) implies
(2) implies
(3) implies
Slutsky's theorems
注意,前面的是随机变量,这里的c是常数
(1) iff
(2) and implies
(3) and implies
那我们怎么用它?引理在此
If and , then
(1)
(2)
(3) If with , and , then
根据这个Slutsky定理我们可以知道t分布
大O
定义
是随机变量
意思是 uniformly tight, 也就是
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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