[Probability Theory_7] Week7

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

$\textbf{Theorem}$

$X_n \xrightarrow{L} X $ and $S_n \xrightarrow{p} 0 \Rightarrow X_n \cdot S_n \xrightarrow{p} 0$

$\textbf{Theorem[Sheffe’s Theorem]}$

Suppose that $\nu_n, \nu$ are measure in $(\mathbb{R}, \mathscr{B}(\mathbb{R}))$

$\nu_n (A) = \int_A f_n d\mu$ $\nu(A) = \int_A f d\mu$ $\mu:$Borel measure and $f_n, f \geq 0$

If $\nu_n(\mathbb{R}) = \nu(\mathbb{R}) < \infty, ^\forall n \geq 1 $ and $\mu \lbrace f_n \nrightarrow f \rbrace = 0$, then $sup_{A \in \mathscr(\mathbb{R})} \vert \nu_n(A) - \nu(A) \vert \leq \int \vert f_n - f_n \vert d\mu \rightarrow 0$ as $n \rightarrow n$

$\textbf{Theorem}$

Let $h:\mathbb{R} \rightarrow \mathbb{R}$, $X_n \xrightarrow{d} X$ , $h(X_n) \Rightarrow h(X), h:$conti. Let $D_h = \lbrace x: h$ is disconti at $ x \rbrace$. If $P(X \in D_h) = 0 \Rightarrow h(X_n) \xrightarrow{d}h(X)$

$\textbf{Theormem[skorohod’s theorem]}$

Suppose that $\mu_n, n\geq 1, \mu$: Borel measure on $(\mathbb{R}, \mathscr{B}(\mathbb{R})) $ s.t. $\mu_n \xrightarrow{w} \mu$. Then $^\exists (\Omega’, \mathcal{F}’, P’) $ and r.v.s $X_n’, X’$ s.t. $X_n’ \sim \mu_n, X’ \sim \mu$ and $X_n’ \rightarrow X’$ $~ P’-a.s.$

$\textbf{Theorem}$

$X_n \xrightarrow{d} X \Leftrightarrow ^\forall$bounded conti ftn $g, Eg(X_n) \rightarrow Eg(X),$ as $n \rightarrow \infty$

$\textbf{Theorem}$

TFAE

1. $X_n \xrightarrow{d} X$
2. $^\forall$open set $G, liminf_nP(X_n \in G) \geq P(X \in G)$
3. $^\forall$Closed set $F, limsup_nP(X_n \in F) \leq P(X \in F)$
4. $^\forall A \in \mathscr{B}(\mathbb{R})$ with $P(X \in \partial A) = 0, \lim_nP(X_n \in A) = P(X \in A)$