[Probability Theory_12] Week12

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

$\textbf{Theorem}$

Let $X_1, X_2, \cdots $ are iid with $E\vert X_1 \vert = \infty$. Let $\lbrace a_n \rbrace $ be a seq of positive numbers s.t. $\dfrac{a_n}{n}$ is monotonly increasing. Then

$\limsup \dfrac{S_n}{a_n} = 0, ~~ \Sigma^\infty_{n=1} P(\vert X_1 \vert \geq a_n) < \infty$
$= \infty, ~~\Sigma^\infty_{n=1} P(\vert X_1 \vert \geq a_n) = \infty$

$\textbf{Theorem}$

Let $X_1, X_2, \cdots $ are iid r.v.s with $E X_1 = 0 , 0 < EX_1^2 = \sigma^2 < \infty$. Let $S_n = X_1 + \cdots + X_n$. Then

$\forall \varepsilon, (n \geq 2), \dfrac{S_n}{\sqrt{n}(\log{n})^{1/2}} \rightarrow 0 ~~a.s.$

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Characteristic Function

$\textbf{Definition}$

Let $\mu$ be a prob Borel measure. Then

$\varphi(t) = \int e^{itx} d\mu(x) = \int \cos tx ~d\mu(x) + i\int \sin tx ~d\mu(x)$

$\textbf{Definition[Ch.f]}$

The Character function of r.v. $X$ is

$\varphi(t) = Ee^{itx} , ~X \sim\mu$