확률론 - 중간고사 대비 정리들 정리

 

 

서론

중간고사를 봐야해요...

중요한 정리들을 정리하겠습니다.

갖다놓고 외워봅시다.

본론

1. Definition of $\pi$, $\lambda$ system

1) $\pi$ system

A collection $P$ is a $\pi$-system if for $A,B \in P$, $A \cap B \in P$

 

2) $\lambda$ system

A collection $L$ is a $\lambda$-system if

- $\emptyset \in L$

- If $A^c \in L$, $A \in L$

- If$A_1,A_2,... \in L$ and $A_1,A_2,...$are disjoint, then $\bigcup_{k=1}^{\infty} A_k \in L$

 

2. $\pi$, $\lambda$ system theorem

Let $P$ and $L$ be $\pi$ and $\lambda$ - system for each.

Then if $P \subset L$, $\sigma (P) \subset L$

 

3. Borel-Cantelli lemma 1,2

Borel-Cantelli lemma 1

If $\sum_{n} P(A_n) < \infty$, then $P(A_n \space i.o.) = 0$

 

pf) $\displaystyle P(A_n \space i.o.) = P(\bigcap_{n=1}^{\infty} \bigcup_{n=1}^{\infty} A_k) = \lim_{n \rightarrow \infty} P(\bigcup_{k=n}^{\infty} A_k) < \lim_{n \rightarrow \infty} \sum_{k=n}^{\infty} P(A_k) = 0$

By countable subadditivity and Tail sum property.(If sum is finite, tail sum converges 0 since tail sum $T_n = S-S_n$)

 

Borel-Cantelli lemma 2

If $A_1, A_2, ... \in \mathcal{F}$ are independent and $\sum_{n=1}^{\infty} P(A_n) = \infty$, then $P(A_n \space i.o.) = 1$ 

 

4. Kolmogorov's extension theorem

Suppose we are given probability measure $\mu_n$ on $(\mathbb{R}^n, R^n)$ that are consistent, that is,

$\mu_{n+1}((a_1,b_1] \times ... \times (a_n,b_n] \times \mathbb{R}$= $\mu_{n}((a_1,b_1] \times ... \times (a_n,b_n]$.

Then there is a unique probability measure $P$ on $(\mathbb{R}^{\mathbb{N}},R^{\mathbb{N}})$ with $P(\omega : \omega_i \in (a_i,b_i], 1 \leq i \leq n) = \mu_n((a_1,b_1] \times ... \times (a_n,b_n])$.

 

marginalization을 하면 그 아래 차원 measure가 됨. 이걸 만족하면 unique하게 확장

 

5. DCT(Dominated Convergence Theorem)

If $X_n \rightarrow X \space a.s.$ and $|X_n| \leq Y$ with $EY<\infty$,

then $EX_n \rightarrow EX \space, n \rightarrow \infty$

 

$X_n$이 almost surely converge하는 건 충분하지 않고 수렴하는 Y가 boundary가 되어야 함.

왜냐면 수렴하지 않는 point들에서 무한대로 방방 뛰어버리면 수렴하지 않을 수 있기 때문

 

6. Fatou's lemma

If $X_n \geq 0 \space a.s.$ and $X_n \rightarrow X \space a.s.$,

then $\displaystyle 0 \leq EX \leq \liminf_{n \rightarrow \infty} EX_n$

 

7. Fubini's theorem

Let $\mu = \mu_x \times \mu_y$ be a probability Borel measure on $(\mathbb{R}^2,B(\mathbb{R}^2))$.

If $h: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a Borel function s.t h is integrable,

then, $\int_{\mathbb{R}^2} h(x,y) d\mu(x,y) = \int_{\mathbb{R}} (\int_{\mathbb{R}} h(x,y) d\mu_x) d\mu_y = \int_{\mathbb{R}} (\int_{\mathbb{R}} h(x,y) d\mu_y) d\mu_x$

 

8. Markov inequality

Let $\phi(x) \geq 0$ be a measurable function and $A \in \mathcal{F}$.

 

Then, $i_A P(X \in A) \leq E[\phi(X) ; X \in A] \leq E[\phi(X)]$ where $i_A = inf{\phi(x) ; x \in A}$.

 

9. Weakly Convergence

A sequence $\mu_n$ of  sub probability borel measure converges weakly to a sub probability borel measure $\mu$,

If $\exists$ a dense subset $D \subset B(\mathbb{R})$ s.t $\space \forall \space a,b \in D \space \mu_n (a,b] \rightarrow \mu(a,b], n \rightarrow \infty$

 

어떤 Dense한 subset에서 half open interval들의 measure가 모두 수렴.

 

사실 $\mu \{a\} = \mu \{b\} = 0$을 만족하는 $a,b$에서 $\mu_n (a,b] \rightarrow \mu(a,b], n \rightarrow \infty$가 만족해도 Okay.

 

10. Scheffé's theorem

Suppose that r.v.s $X_n$ and X have densities $f_n$ and $f \geq 0$ w.r.t a borel measure $\mu$.

 

If $f_n \rightarrow f \space \mu - a.s.$, (density converges $\Rightarrow$ in distribution converges)

 

 

11. Convergence in distribution

$X_n \overset{d}{\longrightarrow} X$ iff $\forall$ every bdd. cont. function $g$ $E[g(X_n)] \overset{n \rightarrow \infty}{\longrightarrow} E[g(X)]$

 

12. Portmanteau Theorem

The following statements are equivalent.

 

1) $X_n \overset{d}{\longrightarrow} X$

2) $\forall$ open $G$   $\liminf P(X_n \in G) \leq P(X \in G)$

3) $\forall$ closed set $F$   $\limsup P(X_n \in F) \leq P(X \in G)$

4) $\forall A \in B(\mathbb{R})$ with $P(\partial A) = 0$  $\displaystyle \lim_{n \rightarrow \infty} P(X \in A)$

 

13. Helly's selection principle

Let $\{F_n\}$ be a seq of dists. Then $\exists$ subseq $\{F_{n_k}\}$ and distribution like $F$ s.t $x \in C(F), F_{n_k}(x) \rightarrow F(x)$

 

14. Tightness

Let $\{F_n\}$ be a seq of dist. Then, every subseq limit is a dist iff $\{F_n\}$ is tight.

 

$\forall \epsilon > 0$ $\exists M>0$  s.t  $P(|X_n| \geq M) = 1 - P(X_n \leq M) + P(X_n \leq -M) = 1- F_n(M) + F_n(-M) \leq \epsilon$

 

15. Kolmogorov's 0-1 law.

note) Tail events let $\mathcal{F}'_n = \sigma(X_n,X_{n+1})...)$.

Then $\mathcal{T} \bigcap_n mathcal{F}'_n$ is called tail $\sigma$-field.

무한히 먼 미래에도 측정 가능한 사건들.

 

If $X_1,X_2,...$ are independent and A $\in \mathcal{T}$ then $P(A) = 0$ or $1$

결론

사람 이름 붙은거 위주로 만들었습니다.

열심히 외웁시다.

 

끗.