[Probability Theory_9] Week9

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

Random Series

$\textbf{Kolmogorov’s inequality}$

Let $\lbrace X_n : n\geq 1 \rbrace$ be a seq of indep r.v.s with $EX_j =0$ and $Var(X_j) = \sigma_j^2 < \infty$. Then $\forall \varepsilon > 0,$

$P(\max_{1\leq j \leq n} \vert S_j \vert \leq \dfrac{\Sigma^n_{j=1} \sigma_j^2}{\varepsilon^2} ), S_j = X_1 + \cdots + X_j$

-n=1이면, Markov’s ineq

$\textbf{Theorem}$

Let $X_n$ be indep r.v.s with $EX_n = 0 $ and $Var(X_n) = \sigma^2n < \infty, S_n = X_1 + \cdots + X_n$.
If $\Sigma^\infty{n=1} \sigma^2n < \infty \Rightarrow \Sigma^\infty{n=1}X_n$ conv a.s.

$\textbf{Theorme[Etermadi’s inequality]}$

$^\forall \varepsilon > 0$,

$P(\max_{1\leq k \leq n} \vert S_k \vert \geq 3\varepsilon ) \leq 3 \max_{1 \leq k \leq n} P(\vert S_k \vert)$

$S_n = X_1 + \cdots + X_n$ (indep)

$\textbf{Theormem[Levy’s Thm]}$

If $S_n \xrightarrow{p} S \Rightarrow S_n \rightarrow S~~ a.s.$
$S_n = X_1 + \cdots + X_n$ (indep)