[Probability Theory_4] Week4

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

$\textbf{Definition[independent]}$

1. $A,B \in \mathcal{F}$ are independent, if $P(A \cap B) = P(A)P(B)$
2. $A_1,A_2, \cdots, A_n$ are independent, if $\forall 1 \leqslant i_1 < \cdots < i_j \leqslant n$, $P(A_{i1} \cap \cdots \cap A_{ij}) = P(A_{i1})\cdots P(A_{ij})$
3. $A_1, A_2, \cdots \in \mathcal{F}$ are independent, if $\forall n \geqslant 1, A_1, \cdots , A_n$ are independent.
   
   

$\textbf{Defininion}$

1. $\mathcal{G}_i \subset \mathcal{F}, i = 1,2, \cdots, n$ are indep, if $\forall A_i \in \mathcal{G}_i, A_1, \cdots , A_n$ are indep
2. $\mathcal{G}_1, \mathcal{G}_2, \cdots \subset \mathcal{F}$ are indep, if $\forall n \geqslant 1, \mathcal{G}_1, \cdots \mathcal{G}_n$ are indep.
   
   

$\textbf{Theormem}$

Assume $\mathcal{G}_i \subset \mathcal{F} , i \geqslant 1$ are indep $\pi$-system, then $\sigma(\mathcal{G}_i) $ are independent.

$\textbf{Theormem[2nd Borel Cantelli’s lemma(theorem)]}$

If $A_1, A_2, \cdots$ are indep and $\Sigma P(A_n) = \infty$, then $P(A_n ~ i.o.)=1$

$\textbf{Definition}$

r.v. $X_1, \cdots, X_n$ are indep, if $\sigma(X_1), \cdots, \sigma(X_n)$ are indep.

Convergence

$\textbf{Definition[conv a.s.]}$

r.v. $X_n$ converges to r.v. $X ~ a.s.$, if $P(X_n \rightarrow X, n \rightarrow \infty) = 1$

remark

If $w \in \cap^\infty_{m=1} \cup^{\infty}{n=1} \cap^{\infty}{k=1} (\vert X_n - X_{n+k} \vert \leqslant 1/m)$

$\rightarrow \forall m, \exists n \geqslant 1$ s.t. $\forall k \geqslant 1, \vert X_n - X_{n+k} \vert \leqslant 1/m$, 즉, $limX_n(w)$ exists

$\textbf{Theorme[Fatou’s lemma]}$

$0 \leqslant X_n \rightarrow X ~ a.s.\Rightarrow EX \leqslant liminf_n EX_n$

$\textbf{Theorme[Monotone Conv Thm, MCT]}$

$0 \leqslant X_n \nearrow X \Rightarrow EX_n \nearrow EX$

$\textbf{Theorme[Dominated Conv Thm, DCT]}$

$X_n \rightarrow X ~ a.s. , \vert X_n \vert \leqslant Y $, with $E \vert Y \vert < \infty $, $\Rightarrow EX_n \rightarrow EX, n \rightarrow \infty$ and $\vert X \vert \leqslant Y$

$\textbf{Theorme[Bounded Conv Thm, DCT]}$

$X_n \rightarrow X ~ a.s. , \vert X_n \vert \leqslant B$(const) $a.s. $ $\Rightarrow EX_n \rightarrow EX, n \rightarrow \infty$

$\textbf{Theorme}$

Assume $X_n \rightarrow X ~ a.s.$ Let $g,h$ be conti ftn with

1. $g \geqslant 0, g(x) > 0, \forall \vert x \vert \geqslant m_0 > 0 $
2. $\dfrac{\vert h(x) \vert }{g(x)} \rightarrow 0 , \vert x \vert \rightarrow \infty$
3. $Eg(X_n) \leqslant K, \forall n \geqslant 1 $
   

Then $Eh(X_n) \rightarrow Eh(X), n \rightarrow \infty$

$\textbf{Difinition}$

Let $(\Omega, \mathcal{F})$, $(\Omega ‘, \mathcal{F}’)$ be two event space and let $\varphi$ be a mapping from $\Omega$ to $\Omega’$. Then $\varphi$ is $\mathcal{F} / \mathcal{F}’$- measurable, if $\forall A \in \mathcal{F}’ , \varphi^{-1}(A) \in \mathcal{F}$

$\textbf{Theorem}$

Set $(\Omega, \mathcal{F}, P)$ be a prob space. and $\varphi$ is $\mathcal{F} / \mathcal{F}’$- measurable. Then for any Borel ftn $f: \Omega’ \rightarrow \mathbb{R}, \int f(\varphi)dP = \int f d(P \varphi^{-1})$