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Math/Analysis

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Signed Measure Signed Measure는 무엇일까? 우리가 이전에 배웠던 measure와 거의 비슷하다! 하지만 여러가지 헷갈리는 것들이 있어서 조심해야 한다. Definition 12.1 in Bass) Let $\mathcal{A}$ be a $\sigma$-algebra. A signed measure is a function $\mu : \mathcal{A}\to (-\infty,\infty]$ such that $\mu(\emptyset)=0$ and if $A_1,A_2,\cdots $ are pairwise disjoint and all the $A_i$ are in $\mathcal{A}$, then $\mu(\cup^{\infty}_{i=1} A_i) = \sum^{\infty}_{i=1} \mu(A..
Limit theorems 여기저기서 많이 쓰이는 Monotonce convergence theroem을 증명해보겠다 Theorem [Bass, Theorem 7.1] Suppose $f_n$ is a sequence of non-negatvie measurable functions with $f_1\leq f_2 \leq \cdots$ with $f_n\to$f in pointwise sense. Then, $\int f_n d\mu \to $\int fd\mu$. Proof $lim_{n\to\infty} \int f_n = L \leq \int f$를 확인하는 것은 쉽다. 이제 $L \geq \int f$라는 것을 보이겠다. $0\leq s \leq f$인 simple function에 대해 $A_n:= \{ x : f_n(..
Sigma-algebra Sigma-algebra의 간단한 예시들을 살펴보겠습니다. Example 2.3) $\mathcal{A}:=\{ A\;\text{or}\;A^c\;\text{is countable}\}$. $ (\cup_i A_i )^c \subset A_{i_0}^c $. 따라서 $A_{i_0}^c$가 countable하므로, $\cup_i A_i \in \mathcal{A}$이다. Example 2.6) Let $X=[0,1]$ and $B_1,\dots,B_8$ be subsets of $X$ which are pariwise disjoint and whose union is all of $X$. Let $\mathcal{A}$ be finite unions of $B_i$'s and empty set. Then ..
Characteristic Function Durrett 책을 일부 정리한 내용입니다. Theorem 3.3.11 ( The inversion formula ) Let $\varphi (t) = \int e^{itx} \mu (dx)$ where $\mu$ is a probability measure. If $a
Weak Law of Large Numbers Theorem 2.2.3 ( L2 weak Law ) Let $X_1, X_2, \dots$ be uncorrelated random variables with $EX_i =\mu$ and $var(X_i)0$ with $b_n\rightarrow \infty$ and let $\bar{X}_{n,k}=X_{n,k}1(|X_{n,k}|\leq b_n)$. Suppose $n\rightarrow\infty$ (i) $\sum\limits^n_{k=1} P( |X_{n,k}|>b_n) \rightarrow 0 $ , and (ii) $b_n^{-2} \sum\limits^n_{k=1} E \bar{X}^2_{n,k} \rightarrow 0 $ If we let $S_n = X_{n,1}+ \dots + X..
Borel-Cantelli Lemma Lemma ( Fatou's Lemma ) (Fatou's Lemma) [1] $$P(\liminf A_n) \leq \liminf P(A_n) \qquad \limsup P(A_n) \leq P(\limsup A_n)$$ Proof) $B_k =\cap_{n\geq k}A_n$ 은 increasing sequence of event이다. 또한 $B_k$의 정의로부터 $B_k \subset A_n$ for any $n$ such that $n\geq k$ 임을 알 수 있다. 즉 $P(B_k) \leq \inf_{n\geq k}P(A_n)$이다. 따라서 $P(\liminf_n A_n) = P( \cup_n B_b) = \lim P(B_n)$ 이다. 여기서 $B_n$이 increasing sequence라는..
Strong Law of Large Numbers With only finite mean we can prove almost sure convergene to mean. proof) Let $Y_k = X_k 1(|X_k|\leq k )$. $T_n = Y_1 + \dots + Y_n$. It is enough to show $T_n/n \rightarrow \mu$ a.s. $\sum\limits^{\infty}_0 P( |X_k| > k ) \leq \int^{\infty}_0 P( |X_1|>t) dt = E|X_1|1$을 생각한다. $\sum\limits^{\infty}_{n=1} P ( |T_{k(n)} - E T_{k(n)} | > \epsilon k(n) )\leq \sum\limits^{\infty}_{n=1} \frac{E|T_{k(n)..
Convergence of Random Series Theorem 2.5.3 ( Kollmogorov's 0-1 law ) If $X_1,\dots$ are independent and $A\in\mathcal{T}$ then $P(A)=0$ or $1$. Proof) (a) $A\in \sigma (X_1,\dots,X_k)$ and $B\in\sigma (X_{k+1},X_{k+2},\dots)$ are independent. Since $\sigma (X_1,\dots,X_k)$ and $\cup_J \sigma (X_{k+1},\dots,X_{k+j})$ are $\pi$ systems. (b) $A\in \sigma (X_1,\dots)$ and $B\in \mathcal{T}$ are independent. $\cup_k \sigma (X_1,..

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