[Probability Theory_11] Week11

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

S.L.L.N

$\textbf{Lemma[Kronecker’s lemma]}$

Assume $\lbrace X_n \rbrace$ is a real seq and $\lbrace a_n \rbrace$ is a positive real seq diverging to $\infty$ monotonely

If $\Sigma \dfrac{x_n}{a_n}$ conv. $\Rightarrow \dfrac{1}{a_n}\Sigma^n_{j=1} X_j \rightarrow 0$

$\textbf{Theorem[S.L.L.N]}$

If $X_1, X_2, \cdots $ are iid with $E\vert X_1 \vert < \infty, EX_1 = \mu$

$\dfrac{X_1 + \cdots +X_n}{n} \rightarrow \mu ~ a.s.$

---

$\textbf{Theorem}$

If $X_i, i\geq1 $are iid with $EX_i^+ = \infty, EX_i^- < \infty$

$\dfrac{S_n}{n} \rightarrow \infty ~ a.s.$

$\textbf{Theorem[Gilvenko-Cantelli theorem]}$

$X_i$ iid r.v.s $\sim F$. Then $F_n(x) \rightarrow F(x) ~ a.s.$

$\textbf{Theorem}$

If $X_i, i \geq 1$ are iid with $E\vert X_1 \vert = \infty$, then

$P(\vert X_n \vert \geq n ~~~i.o.) = 1$ and $P(\lim_n \dfrac{S_n}{n}$ exists & $ \in (-\infty, \infty) ) =0$