[Probability Theory_13] Week13

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

$\textbf{Inversion formula}$

Let $F$ be a distn ftn and $\varphi(t) = \int e^{itx} dF(x)$. Then $^\forall x_1 < x_2$

$\dfrac{F(x_2)-F(x_2-)}{2} - \dfrac{F(x_1)-F(x_1-)}{2} = \lim_{T\rightarrow \infty}\dfrac{1}{2\pi}\int ^T_T \dfrac{e^{-itx_1} - e^{-itx_2}}{it} g(t) dt$

$\textbf{Theorem}$

Assume $\int^\infty_\infty \vert g(t) \vert dt < \infty $, $\varphi(t) = \int e^{itx} dF(x)$. Then

$F$ is differentiable, and $F'(x) = \dfrac{1}{2\pi} \int^\infty_\infty e^{-ixt} \varphi(t)dt$

$\textbf{Theorem}$

For each $x_0$, $\lim_{T \rightarrow \infty} \dfrac{1}{2T} \int^T_T e^{-itx_0} \varphi(t) = \mu \lbrace x_0 \rbrace$

$\textbf{Theorem}$

$\lim_{T \rightarrow \infty} \dfrac{1}{2T} \int^T_T \vert g(t) \vert ^2 dt = (\Sigma_x\mu \lbrace x \rbrace)^2$

$\textbf{Theorem}$

$F’ = f \Rightarrow \varphi(t) \rightarrow 0 , t \rightarrow \infty$