[Probability Theory_10] Week10

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

Three Series Theorem

$\textbf{Lemma}$

Suppose that $X_n, n\geq 1$ are indep with finite variance. If $E\vert X_n - EX_n \vert \leq A ~ a.s. ^\forall n,\varepsilon$

$P(\max_{1\leq j \leq n} \vert S_j \vert \leq \varepsilon ) \leq \dfrac{(2A + 2\varepsilon)^2}{Var(S_n)} $

$S_n = X_1 + \cdots + X_n$

$\textbf{Definition}$

$\lbrace X_n \rbrace, \lbrace Y_n \rbrace$ are equivalent(eventually), if $\Sigma^\infty_{n=1} P(X_n \ne Y_n) < \infty$

$\textbf{Theorem[Three Series Theorem]}$

Let $\lbrace X_n \rbrace$ be a seq of indep r.v.s and for $A>0$, Set $Y_n = X_n \cdot I(\vert X_n \vert \leq A)$. Then
$\Sigma X_n$ conv a.s $\Leftrightarrow$

1. $\Sigma^\infty_{n=1} P(\vert X_n \vert > A) < \infty$
2. $\Sigma^\infty_{n=1} EY_n < \infty$
3. $\Sigma^\infty_{n=1} Var(Y_n) < \infty$

$\textbf{Lemma}$

Assume $\lbrace X_n \rbrace \sim \lbrace Y_n \rbrace$ and $a_n \rightarrow \infty (a_n \ne 0)$. Then

$\dfrac{1}{a_n} \Sigma^n_{j=1} X_j \xrightarrow{p} Z \Leftrightarrow \dfrac{1}{a_n} \Sigma^n_{j=1} Y_j \xrightarrow{p} Z$

$(X_j(w) = Y_j(w), j \geq N(w))$

W.L.L.N

$\textbf{Weak law of large numbers (W.L.L.N)}$

$X_1, X_2, \cdots$ are iid with $EX_i =\mu$ $\Rightarrow \dfrac{1}{n} \Sigma^n_{j=1}X_j \xrightarrow{p} \mu$

$\textbf{Lemma}$

$\Sigma^\infty_{n=1} P(\vert X \vert \geq n) \leq E\vert X \vert \leq 1 + \Sigma^\infty_{n=1} P(\vert X \vert \geq n)$

$\Sigma^\infty_{n=1} P(\vert X \vert \geq n) < \infty \Leftrightarrow E\vert X \vert < \infty$