The \(\sigma\)-field generated by a random variable \(X\), denoted \(\sigma(X)\) is the smallest \(\sigma\)-field w.r.t which \(X\) is measurable (as a mapping into (\(\mathbb{R},\mathcal{R}\))).
Similarly, the \(\sigma\)-field generated by a random vector \(X=(X_1,\ldots,X_k)\), again denoted \(\sigma(X)\) is the smallest \(\sigma\)-field w.r.t which \(X\) is measurable (as a mapping into (\(\mathbb{R}^k,\mathcal{R}^k\))).
Finally, the \(\sigma\)-field generated by an arbitrary collection of random variables \(\{X_t,t\in T\}\)(defined on a common probability space \((\Omega, \mathcal{F},P)\)), is the smallest \(\sigma\)-field w.r.t which all \(X_t,t\in T\) are measurable. This \(\sigma\)-filed is denoted \(\sigma(X_t,t\in T)\).
Let \(X=(X_1,\ldots,X_k)\) be a random vector. Then
\(\sigma(X)=\sigma(X_1,\ldots,X_k)=\{ X^{-1}(H):H\in \mathcal{R}^k\}\).
A random variable \(Y\) is \(\sigma(X)\)-measurable if and only if \(Y=f(X)\) for some Borel measurable function \(f:\mathbb{R}^k\rightarrow \mathbb{R}\).
For a random variable \(X\),
\[
\sigma(X)=\sigma \left(\left\{\omega:X(\omega)\le x, x\in \mathbb{R} \right\} \right)=\sigma\left(\left\{X^{-1}((-\infty,x]), x\in \mathbb{R} \right\} \right).
\]
중요하다.
Random variables (random vectors) \(X_1,\ldots X_k\) are independent if the \(\sigma\)-fields \(\sigma(X_1),\ldots,\sigma(X_k)\) are independent, or equivalently, if \[
P(X_1\in H_1,\ldots X_k\in H_k)= P(X_1\in H_1)\cdots P(X_k\in H_k) \mbox{ }\mbox{ }\mbox{ for all }H_1,\ldots,H_k\in \mathcal{R}^1.
\]
Set과 Class들에 대한 얘기에서 Random variable과 \(\sigma\)-fields of random variable에 대한 스토리로 바뀌었다.
내용은 전부 같다.
Random variables \(X_1,\ldots,X_k\) are independent if and only if \[
\mu=\mu_1\times\cdots\times\mu_k, \mbox{ }\mbox{ }\mbox{ (product measure)},
\] or equivalently, \[
F(x)=F_1(x_1)\cdots F_k(x_k)\mbox{ }\mbox{ }\mbox{ for all } x=(x_1,\ldots,x_k)\in \mathbb{R}^k.
\]
If \(X_1,\ldots,X_k\) are independent random variables and \(g_1,\ldots,g_k\) are Borel measurable functions, then \(g_1(X_1),\ldots,g_k(X_k)\) are independent random variables.
\(g_i(X_i)\) is measurable w.r.t the corresponding \(\sigma(X_i)\).
즉, \(\omega\)의 관점에서 생각할 때 \(g_i(X_i)\in \sigma(X_i)\implies\sigma(g_i(X_i))\subset \sigma(X_i)\), \(i=1,\ldots,k\) (\(g_i(X_i\)는 합성함수이기 때문).
Because \(\sigma(X_1),\ldots,\sigma(X_k)\) are independent, it follows immediately from the definition of independent classes that \(\sigma(g_1(X_1)),\ldots,\sigma(g_k(X_k))\) are also independent
\((g_1(X_1),\ldots,g_k(X_k)\)은 choices in \(\sigma(g_1(X_1)),\ldots,\sigma(g_k(X_k))\implies\) choices in \(\sigma(X_1),\ldots,\sigma(X_k))\).
If \(X\) and \(Y\) are independent random variables, either both nonnegative or both integrable, then \[ E(XY)=E(X)E(Y). \] * 이 Theorem은 기대값을 split하는 것뿐만이 장점이 아니라, \(X,Y \in L^1\)이고 \(X,Y\) 가 독립이면 \(XY\in L^1\)임을 보인다는 것이 놀랍다.
그동안 \(XY\in L^1\)를 보이기 위해서는 Holder’s inequality나 Cauchy-Schwarz inequality에서처럼 \(p,q\ge 1\)이고 \(1/p+1/q=1\)을 만족하는 \(X\in L^p\), \(Y\in L^q\)가 요구되었다.
\(X,Y\) : nonnegative simple r.v \(\rightarrow\) nonnegative r.v \(\rightarrow\) general integrable r.v 순서로 증명한다.
중요하다
Suppose that \(X\) and \(Y\) are independent random vectors (\(k\) and \(m\) dimensional, respectively) with respective distributions \(P_X\) and \(P_Y\). Let \(g:\mathbb{R}^{k+m}\rightarrow \mathbb{R}\) be a Borel measurable function, and let \(A\in \mathcal{R}^m\). if either \(g\) is nonnegative, or \(g(X,Y)\) is integrable, then \[ E[g(X,Y)I_A(Y)]=\int_A E[g(X,y)]dP_Y(y). \] * 증명 : \(X, Y\)가 독립이기 때문에 이 챕터의 4번째 theorem에 의해 joint distribution은 product of each distribution(product measure)이다. 그러므로 change of variable theorem과 Fubini’s theorem에 의해
\[\begin{eqnarray*} E[g(X,Y)I_A(Y)] &=& \int_{\Omega_1\times\Omega_2}g(X(\omega_1),Y(\omega_2))I_{Y^{-1}(A)}(\omega)\mbox{ }dP(\omega_1 \times \omega_2) \\ &=&\int_{\mathbb{R}^{k+m}}g(x,y)I_A(y)\mbox{ }dP_X \times P_Y(x \times y)\\ &=&\int_{\mathbb{R}^{m}}\int_{\mathbb{R}^{k}}g(x,y)I_A(y)\mbox{ }d P_X(x)P_Y(y)\\ &=&\int_{\mathbb{R}^{m}}I_A(y)\left[\int_{\mathbb{R}^{k}}g(x,y)\mbox{ }d P_X(x)\right]P_Y(y)\\ &=&\int_{A}E\left[g(X,y)\right]P_Y(y)\\ \end{eqnarray*}\]