[Probability Theory_5] Week5

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

$\textbf{Definition}$

Let $(\Omega, \mathcal{F}, P)$ be a prob space.

1. $\mathcal{L}_0 =$ the clss of all r.v.s on $(\Omega, \mathcal{F})$
2. $\mathcal{L}_p = \lbrace X \in \mathcal{L}_0 : E \mid X \mid ^p < \infty \rbrace $
3. $\mathcal{L}_\infty = \lbrace X \in \mathcal{L}_0 : \mid X \mid < M \ \ a.s.$ for some $ M > 0 \rbrace $

$0<p<\infty : \Vert X \Vert_p = (E \mid X \mid ^p)^{1/p}$ (norm)
$\ \ \ \ \ \ \ p=\infty : \Vert X \Vert_\infty = ess~sup \mid X \mid = inf \lbrace M > 0 : \mid X \mid \leqslant M ~ a.s. \rbrace $

• $p=2$인 경우, $\mathcal{L}_2$는 Hilbert space
  
  

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Inequality

$\textbf{Markov’s Inequality}$

$P(\mid X \mid \geqslant C) \leqslant \dfrac{E\mid X\mid}{C}$

$\textbf{Triangle Inequality}$

$ X_i \in \mathcal{L}_p, p \geqslant 1,$
$ \Vert X_1 \Vert \leqslant \Vert X_1 \Vert + \cdots + \Vert X_n \Vert$

$\textbf{Holder’s Inequality}$

$X \in \mathcal{L}_p$ and $Y \in \mathcal{L}_q$ and $1/p+1/q =1, p,q>1$
$E\vert XY\vert \leq \Vert X \Vert_p\Vert Y \Vert_q $

$\textbf{Gensen’s Inequality}$

$\varphi: \mathbb{R} \rightarrow \mathbb{R}$ is convex
$\varphi(EX) \leq E\varphi(X)$

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$\textbf{Theorem}$

If $X \geq 0, EX = \int^{\infty}_0 P(X > t)dt$

$\textbf{Theorem}$

Suppose that $X, Y$ are indep with dist $\mu$ and $\nu$. If $h: \mathbb{R}^2 \rightarrow \mathbb{R}$: a Borel ftn with $h \geq 0$ or $E \vert h(X,Y) \vert < \infty$. Then
$Eh(X,Y) = \int\int h(x,y) \mu(dx) \nu(dy) = \int\int h(x,y) \nu(dy) \mu(dx)$

Tail event

$\textbf{Tail event}$

Let $A_i \in \mathcal{F}, n \geq 1.$
The tail $\sigma$-field of $\lbrace A_n \rbrace$ is $\tau = \cap^{\infty}{n=1} \sigma(A_n, A{n+1}, \cdots) : \sigma$-field

$\textbf{Theorme[Kolmogorov’s 0-1 law]}$

Assume $A_n$ are independent
If $A \in \tau , P(A) = 0$ or $1$

$\textbf{Definition}$

Let $X_n$ be r.v.s and
let $\mathcal{G}n = \sigma(X_n, X{n+1}, \cdots) = \sigma\lbrace (X_i \in B): B \in \mathscr{B}(\mathbb{R}), i=n,n+1,\cdots \rbrace$
$\tau = \cap^{\infty}_{n=1} \mathcal{G}_n :$ tail $\sigma$ - field of $\lbrace X_n \rbrace$

$\textbf{Difinition}$

1. $\lbrace X_n \rbrace$ converges to $X$ in probability, if $\forall \varepsilon > 0 , P(\vert X_n - X \vert \geq \varepsilon) \rightarrow 0, n \rightarrow \infty$
2. $\lbrace X_n \rbrace$ converges to $X ~a.s.$, if $\exists C \in \mathcal{F}$ with $P(C) =1 $ s.t. $\forall w \in C, X_n(w) \rightarrow X(w), n \rightarrow \infty$
   
   

$\textbf{Theorme}$

$\lbrace X_n \rbrace \rightarrow X ~a.s.$
$\Leftrightarrow \forall \varepsilon, P(\cup^\infty_{k=n} \vert X_k - X \vert > \varepsilon) \rightarrow 0, n \rightarrow \infty$
$\Leftrightarrow \forall m \geq1 , P(\cup^\infty_{k=n} \vert X_k - X \vert > 1/m) \rightarrow 0, n \rightarrow \infty$
$\Leftrightarrow \forall m \geq1 , P(\cap^\infty_{n=1} \cup^\infty_{k=n} \vert X_k - X \vert > 1/m) = 0, n \rightarrow \infty$
$\Leftrightarrow \forall m \geq1 , P(\vert X_n - X \vert > 1/m ~ i.o.) \rightarrow 0$

$\textbf{Theorme}$

$f: \mathbb{R} \rightarrow \mathbb{R}$ is conti
$X_n \xrightarrow{p}X \Rightarrow f(X_n) \xrightarrow{p} f(X)$