[Probability Theory2] Week2

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

$\pi - \lambda~$ System Theorem

$\textbf{Proposition}$

Let $\mathscr{L}$ be any $\lambda$-system then $\forall A \in \mathscr{L}, \mathcal{G}_A = \lbrace B \subset \Omega: A\cap B \in \mathscr{L}$ is $\lambda$-system $\rbrace$

$\textbf{Theorem}$

Let $\mathscr{P},\mathscr{L}$ be $\pi, \lambda$-system. If $\mathscr{P}\subset \mathscr{L}, \rightarrow \sigma(\mathscr{P}) \subset \mathscr{L}$

$\textbf{Theorem}$

Let $(\Omega, \mathcal{F})$ be an event space. If $P, Q$ are two prob measure and $P(A) = P(Q),~ \forall A \in \mathscr{P} \subset \mathcal{F}$ where $\mathscr{P}$ is a $\pi$-sys. Then $P=Q$ on $\sigma( \mathscr{P})$

Borel-Cantelli lemma

$\textbf{Definition}$

Let $A_i \in \mathcal{F}, i \geqslant 1$ Set

$\cap^{\infty}_{n=1}( \cup^{\infty}_{k=n}A_k) = limsup_n A_n$
$\cup^{\infty}_{n=1}( \cap^{\infty}_{k=n}A_k) = liminf_n A_n$

$\textbf{Borel-Cantelli lemma}$

if $ \Sigma^{\infty}_{n=1} P(A_n) < \infty \rightarrow P(limsup_nAn)=0 $

• note that $P(limsup_nAn) = P(\cap^{\infty}{n=1} \cup^{\infty}{k=n}A_k) = P(A_n ~ i.o.)$

Random Variable

$\textbf{Definition[random variable]}$

Assume $X$ is a mapping from $\Omega$ to $\mathbb{R}$. and $X$ is a random variable if $\forall B \in \mathscr{B}(\mathbb{R}), (X \in B) = X^{-1}(B) = \lbrace w \in \Omega : X(w) \in B \rbrace \in \mathcal{F}$

• note that $X:(\Omega, \mathcal{F}) \mapsto (\mathbb{R}, \mathscr{B}(\mathbb{R})) $

$\textbf{Theorem}$

$X$ is a r.v. if $\forall B \in \mathscr{C},$ with $\sigma(\mathscr{R}) = \mathscr{B}(\mathbb{R}), (X \in B)\in \mathcal{F}$

$\textbf{Definition[random vector]}$

$X = (X_1, \cdots, X_d)(d \geqslant 2)$ is a random vector if each $X_i$ is a r.v.

$\textbf{Definition[random vector]}$

$X$ is a random vector if $\forall B \in \mathscr{B}(\mathbb{R^d}), (X \in B) \in \mathcal{F}$