[Probability Theory_8] Week8

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

$\textbf{Lemma[Diagonal process]}$

Let $\lbrace x_j: j=1,2, \cdots \rbrace$ be a double array of real numbers. s.t. $\vert x_{ij} \leq b_i \vert, ^\forall j$. Then $^\exists$an increasing seq $n_k, k\geq 1 (n_k \uparrow \infty )$ s.t. $^\forall r =1,2, \cdots, X_{rn_k}, k \geq 1 $ converges

$\textbf{Helly’s Selection Principle}$

Let $\lbrace F_n \rbrace$ be a seq of dists. Then $\exists$subseq $\lbrace F_{n_k}: k\geq 1 \rbrace \subset \lbrace F_n \rbrace$ and a dist-like $F$ s.t. $\forall x \in C(F), F_{n_k}(x) \rightarrow F(x), k \rightarrow \infty$

$\textbf{Theorem}$

Let $F_n$ be a seq of dist. Then every subseq limit is a dist $\text{if and only if}$ $\lbrace F_n \rbrace$ is tight. i.e. $^\forall \varepsilon >0, ^\exists M>0 $ s.t. $ 1 - F_n(M) + F_n (-M) \leq \varepsilon, ^\forall n \geq 1$