[Probability Theory_6] Week6

서울대학교 통계학과 대학원 2021년도 1학년 1학기 확률론1 수업의 정리내용입니다.

reference: Lecture note (Prof.sangSangyeol Lee),
Probability: Theory and Examples, Rick Durrett, Version 5 January 11, 2019

Expected Value

$\textbf{Theorem}$

Assume $X_n \xrightarrow{p} X$. Then
$\exists subseq \lbrace X_{n’} \rbrace \subset \lbrace X_{n} \rbrace$ s.t. $X_{n’} \rightarrow X ~ a.s.$

$\textbf{Theorem[Fatou’s lemma and DCT]}$

1. $X_n \geq 0, X_n \xrightarrow{p} X \Rightarrow EX \leq liminf_n EX_n$
2. $ X_n \xrightarrow{p} X $ and $\vert X_n \vert \leq Y$ and $EY < \infty \Rightarrow EX_n \rightarrow EX$
   
   

$\textbf{Definition}$

A sequence $\lbrace \mu_n : n \geq 1 \rbrace$ of sub porb $(\mathbb{R}, \mathscr{B}(\mathbb{R}))$
Borel measures weakly converges to sub porb mea’ $\mu$
,if $^\exists$ dense subset $D \subset \mathbb{R} (\bar{D} \subset \mathbb{R})$ s.t. $^\forall a,b, \in D, \mu_n(a,b] \rightarrow \mu(a,b]$ as $n \rightarrow \infty$

$\textbf{Theorem}$

TFAE

1. $\mu_n \xrightarrow{w} \mu$
2. $^\forall (a,b)$ and $\varepsilon > 0 $,
   $\mu(a+\varepsilon , b -\varepsilon) -\varepsilon \leq \mu_n(a,b] \leq \mu(a - \varepsilon, b+ \varepsilon) + \varepsilon, ~ ^\forall large \ \ n$
3. $^\forall (a,b]$ with $\mu\lbrace a\rbrace = \mu\lbrace b\rbrace = 0$,
   $ \mu_n(a,b] \rightarrow \mu (a,b], n\rightarrow \infty $
   
   

$\textbf{Definition}$

Let $\lbrace F_n, n \geq 1 \rbrace$ be a seq of distribution and let $F$ be a distribution. Then $F_n$ weakly converges to $F$, if $\mu_n \xrightarrow{w} \mu,$ where $ \mu_n \sim F_n, \mu \sim F $

$\textbf{Theorem}$

$F_n \Rightarrow F$ iff
$^\forall x \in C(F) = \lbrace x: F $ is conti at $x \rbrace$, $F_n(x) \rightarrow F(x), n \rightarrow \infty$

$\textbf{Definition}$

$X_n$ converges to $X$ in distribution if
$\mu_n \xrightarrow{w} \mu, ~ ~ X_n \sim \mu_n, X \sim \mu$

$\textbf{Theorme}$

$X_n \xrightarrow{p} X \Rightarrow X_n \xrightarrow{d} X$