lecture 14 -> 15
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@ -46,7 +46,7 @@ First, let us recall some basic definitions:
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\end{fact}
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The converse to this fact is also true:
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\begin{theorem}[Kolmogorov's existence theorem / basic existence theorem of probability theory]
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\label{kolmogorovxistence}
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\label{kolmogorovexistence}
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Let $\cF(\R)$ be the set of all distribution functions on $\R$
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and let $\cM(\R)$ be the set of all probability measures on $\R$.
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Then there is a one-to-one correspondence between $\cF(\R)$ and $\cM(\R)$
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@ -1,255 +1,162 @@
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\lecture{14}{2023-06-06}{}
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\lecture{14}{2023-05-25}{Conditional expectation}
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We want to derive some properties of conditional expectation.
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\section{Conditional expectation}
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\begin{theorem}[Law of total expectation] % Thm 1
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\label{ceprop1}
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\label{totalexpectation}
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\[
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\bE[\bE[X | \cG ]] = \bE[X].
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\]
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\end{theorem}
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\begin{proof}
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Apply (b) from the definition for $G = \Omega \in \cG$.
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\end{proof}
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\begin{theorem} % Thm 2
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\label{ceprop2}
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If $X$ is $\cG$-measurable, then $X = \bE[X | \cG]$ a.s..
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\end{theorem}
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\begin{proof}
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Suppose $\bP[X \neq Y] > 0$.
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Without loss of generality $\bP[X > Y] > 0$.
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Hence $\bP[ X > Y + \frac{1}{n}]> 0$ for some $n \in \N$.
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Let $A \coloneqq \{X > Y + \frac{1}{n}\}$.
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% TODO
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\end{proof}
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\subsection{Introduction}
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\begin{example}
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Suppose $X \in L^1(\bP)$, $\cG \coloneqq \sigma(X)$.
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Then $X$ is measurable with respect to $\cG$.
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Hence $\bE[X | \cG] = X$.
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\end{example}
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\begin{theorem}[Linearity]
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\label{ceprop3}
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\label{celinearity}
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For all $a,b \in \R$
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we have
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\[
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\bE[a X_1 + bX_2 | \cG] = a \bE[X_1 | \cG] + b \bE[X_2|\cG].
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\]
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\end{theorem}
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\begin{proof}
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Trivial % TODO
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\end{proof}
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\begin{theorem}[Positivity]
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\label{ceprop4}
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% 4
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\label{cpositivity}
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If $X \ge 0$, then $\bE[X | \cG] \ge 0$ a.s.
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\end{theorem}
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\begin{proof}
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Let $W $ be a version of $\E[X | \cG]$.
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Suppose $\bP[ W < 0] > 0$.
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Then $G \coloneqq \{W < -\frac{1}{n}\} \in \cG$
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For some $n \in \N$, we have $\bP[G] > 0$.
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However it follows that
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\[
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\int_G \bP[X | \cG] \dif \bP \le -\frac{1}{n} \bP[G] < 0 \le \int_G X \dif \bP.
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\]
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\end{proof}
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\begin{theorem}[Conditional monotone convergence theorem]
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\label{ceprop5}
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% 5
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\label{mcmt}
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Let $X_n,X \in L^1(\Omega, \cF, \bP)$.
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Suppose $X_n \ge 0$ with $X_n \uparrow X$.
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Then $\bE[X_n|\cG] \uparrow \bE[X|\cG]$.
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\end{theorem}
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\begin{proof}
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Let $Z_n$ be a version of $\bE[X_n | Y]$.
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Since $X_n \ge 0$ and $X_n \uparrow$,
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by \autoref{cpositivity},
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we have
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\[
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\bE[X_n | \cG] \overset{\text{a.s.}}{\ge } 0
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\]
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and
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\[
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\bE[X_n | \cG] \uparrow \text{a.s.}
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\]
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(consider $X_{n+1} - X_n$ ).
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Define $Z \coloneqq \limsup_{n \to \infty} Z_n$.
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Then $Z$ is $\cG$-measurable
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and $Z_n \uparrow Z$ a.s.
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Take some $G \in \cG$.
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We know by (b) % TODO REF
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that $\be[Z_n \One_G] = \bE[X_n \One_G]$.
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The LHS increases to $\bE[Z \One_G]$ by the monotone
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convergence theorem.
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Again by MCT, $\bE[X_n \One_G]$ increases to
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$\bE[X \One_G]$.
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Hence $Z$ is a version of $\bE[X | \cG]$.
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\end{proof}
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\begin{theorem}[Conditional Fatou]
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\label{ceprop6}
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\label{cfatou}
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Let $X_n \in L^1(\Omega, \cF, \bP)$, $X_n \ge 0$.
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Then
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\[
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\bE[ \liminf_{n \to \infty} X_n | \cG] \le \liminf_{n \to \infty} \bE[X_n | \cG].
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\]
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\end{theorem}
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\begin{proof}
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\todo{in the notes}
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\end{proof}
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\begin{theorem}[Conditional dominated convergence theorem]
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\label{ceprop7}
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\label{cdct}
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Let $X_n,X \in L^1(\Omega, \cF, \bP)$.
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Suppose $|X_n(\omega)| < X(\omega)$ a.e.~
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and $\int |X| \dif \bP < \infty$.
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Then $X_n(\omega) \to X\left( \omega \right) \implies \bE[ X_n | \cG] \to \bE[X | \cG]$.
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\end{theorem}
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\begin{proof}
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\todo{in the notes}
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\end{proof}
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Recall
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\begin{theorem}[Jensen's inequality]
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If $c : \R \to \R$ is convex and $\bE[|c \circ X|] < \infty$,
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then $\bE[c \circ X] \ge c(\bE[X])$.
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\end{theorem}
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For conditional expectation, we have
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\begin{theorem}[Conditional Jensen's inequality]
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\label{ceprop8}
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\label{cjensen}
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Let $X \in L^1(\Omega, \cF, \bP)$.
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If $c : \R \to \R$ is convex and $\bE[|c \circ X|] < \infty$,
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then $\bE[c \circ X | \cG] \ge c(\bE[X | \cG])$ a.s.
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\end{theorem}
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\begin{fact}
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\label{convapprox}
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If $c$ is convex, then there are two sequences of real numbers
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$a_n, b_n \in \R$
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such that
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\[
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c(x) = \sup_n(a_n x + b_n).
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\]
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\end{fact}
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\begin{refproof}{cjensen}
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By \autoref{convapprox}, $c(x) \ge a_n X + b_n$
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for all $n$.
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Hence
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\[
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\bE[c(X) | \cG] \ge a_n \bE[X | \cG] + \bE[b_n | \cG]
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= a_n \bE[X | \cG] + b_n \text{a.s.}
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\]
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for all $n$.
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Using that a countable union of sets o f measure zero has measure zero,
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we conclude that a.s~this happens simultaneously for all $n$.
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Hence
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\[
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\bE[c(X) | \cG] \ge \sup_n (a_n \bE[X | \cG] + b_n) \overset{\text{\autoref{convapprox}}}{=} c(\bE(X | \cG)).
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\]
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\end{refproof}
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Recall
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\begin{theorem}[Hölder's inequality]
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Let $p,q \ge 1$ such that $\frac{1}{p} + \frac{1}{q} = 1$.
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Suppose $X \in L^p(\bP)$ and $Y \in L^q(\bP)$.
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Then
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\[
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\bE(X Y) \le \underbrace{\bE(|X|^p)^{\frac{1}{p}}}_{\text{\reflectbox{$\coloneqq$}} \|X\|_{L^p}} \bE(|Y|^q)^{\frac{1}{q}}.
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\]
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\end{theorem}
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\begin{theorem}[Conditional Hölder's inequality]
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\label{ceprop9}
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\label{choelder}
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Let $p,q \ge 1$ such that $\frac{1}{p} + \frac{1}{q} = 1$.
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Suppose $X \in L^p(\bP)$ and $Y \in L^q(\bP)$.
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Then
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\[
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\bE(X Y | \cG) \le \bE(|X|^p | \cG)^{\frac{1}{p}} \bE(|Y|^q | \cG)^{\frac{1}{q}}.
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\]
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\end{theorem}
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\begin{proof}
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Similar to the proof of Hölder's inequality.
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\todo{Exercise}
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\end{proof}
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\begin{theorem}[Tower property]
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% 10
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\label{ceprop10}
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\label{ctower}
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Suppose $\cF \supset \cG \supset \cH$ are sub-$\sigma$-algebras.
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Then
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\[
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\bE\left[\bE[X | \cG] \mid \cH\right] = \bE[X | \cH].
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\]
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\end{theorem}
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\begin{proof}
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\todo{Exercise}
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\end{proof}
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\begin{theorem}[Taking out what is known]
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% 11
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\label{ceprop11}
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\label{takingoutwhatisknown}
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If $Y$ is $\cG$-measurable and bounded, then
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\[
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\bE[YX| \cG] \overset{\text{a.s.}}{=} Y \bE[X | \cG].
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\]
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\end{theorem}
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\begin{proof}
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Assume w.l.o.g.~$X \ge 0$.
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Assume $Y = \One_B$, then $Y$ simple, then take the limit (using that $Y$ is bounded).
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\todo{Exercise}
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\end{proof}
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Consider a probability space $(\Omega, \cF, \bP)$
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and two events $A, B \in \cF$ with $\bP(B) > 0$.
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\begin{definition}
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Let $\cG$ and $\cH$ be $\sigma$-algebras.
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We call $\cG$ and $\cH$ \vocab[$\sigma$-algebra!independent]{independent},
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if % TODO
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The \vocab{conditional probability} of $A$ given $B$ is defined as
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\[
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\bP(A | B) \coloneqq \frac{\bP(A \cap B)}{\bP(B)}.
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\]
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\end{definition}
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\begin{theorem}[Role of independence]
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\label{ceprop12}
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\label{roleofindependence}
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If $\cH$ is a sub-$\sigma$-algebra of $\cF$ and $\cH$ is independent
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of $\sigma(\sigma(X), \cG)$, then
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Suppose we have two random variables $X$ and $Y$ on $\Omega$,
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such that $X$ takes distinct values $x_1, x_2,\ldots, x_{m}$
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and $Y$ takes distinct values $y_1,\ldots, y_n$.
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Then for this case, define the \vocab{conditional expectation}
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of $X$ given $Y = y_j$ as
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\[
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\bE[X | \sigma(\cG, \cH)] \overset{\text{a.s.}}{=} \bE[X | \cG].
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\bE[X | Y = y_j] \coloneqq \sum_{i=1}^m x_i \bP[X=x_i | Y = y_j].
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\]
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\end{theorem}
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\begin{example}
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If $X$ is independent of $\cG$,
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then $\bE[X | \cG] \overset{\text{a.s.}}{=} \bE[X]$.
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\end{example}
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\begin{example}[Martingale property of the simple random walk]
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Suppose $X_1,X_2,\ldots$ are i.i.d.~with $\bP[X_i = 1] = \bP[X_i = -1] = \frac{1}{2}$.
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Let $S_n \coloneqq \sum_{i=1}^n X_i$ be the \vocab{simple random walk}.
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Let $\cF$ denote the $\sigma$-algebra on the product space.
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Define $\cF_n \coloneqq \sigma(X_1,\ldots)$.
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Intuitively, $\cF_n$ contains all the information gathered until time $n$.
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We have $\cF_1 \subset \cF_2 \subset \cF_3 \subset \ldots$
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For $\bE[S_{n+1} | \cF_n]$ we obtain
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The random variable $Z = \bE[X | Y]$
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is defined as follows:
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If $Y(\omega) = y_j$ then
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\[
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Z(\omega) \coloneqq \underbrace{\bE[X | Y = y_j]}_{\text{\reflectbox{$\coloneqq$}} z_j}.
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\]
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Note that $\Omega_j \coloneqq \{\omega : Y(\omega) = y_j\}$
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defines a partition of $\Omega$ and on each $\Omega_j$
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(``the $j^{\text{th}}$ $Y$-atom'')
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$ Z$ is constant.
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Let $\cG \coloneqq \sigma(Y)$.
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Then $Z$ is measurable with respect to $\cG$.
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Furthermore
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\begin{IEEEeqnarray*}{rCl}
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\bE[S_{n+1} | \cF_n] &\overset{\autoref{celinearity}}{=}&
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\bE[S_n | \cF_n] + \bE[X_{n+1} | \cF_n]\\
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&\overset{\text{a.s.}}{=}& S_n + \bE[X_{n+1} | \cF_n]\\
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&\overset{\text{\autoref{ceprop12}}}{=}& S_{n} + \bE[X_n]\\
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&=& S_n
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\int_{\{Y = y_j\} } Z \dif \bP &=& z_j \int_{\{Y = y_j\}} \dif \bP\\
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&=& z_j \bP[Y=y_j]\\
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&=&\sum_{i=1}^m x_i \bP[X = x_i | Y = y_j] \bP[Y = y_j]\\
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&=&\sum_{i=1}^m x_i \bP[X = x_i, Y = y_j]\\
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&=& \int_{\{Y = y_j\}} X \dif \bP.
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\end{IEEEeqnarray*}
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Hence
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\[
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\int_{G} Z \dif \bP = \int_{G} X \dif \bP
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\]
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for all $G \in \cG$.
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\end{example}
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We now want to generalize this to arbitrary random variables.
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\begin{theorem}
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\label{conditionalexpectation}
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Let $(\Omega, \cF, \bP)$ be a probability space, $X \in L^1(\bP)$
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and $\cG \subseteq \cF$ a sub-$\sigma$-algebra.
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Then there exists a random variable $Z$
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such that
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\begin{enumerate}[(a)]
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\item $Z$ is $\cG$-measurable and $Z \in L^1(\bP)$,
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\item $\int_G Z \dif \bP = \int_G X \dif \bP$
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for all $G \in \cG$.
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\end{enumerate}
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Such a $Z$ is unique up to sets of measure $0$ and is
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called the \vocab{conditional expectation} of $X$ given
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the $\sigma$-algebra $\cG$ and written
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$Z = \bE[X | \cG]$.
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\end{theorem}
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\begin{remark}
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Suppose $\cG = \{\emptyset, \Omega\}$,
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then
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\[
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\bE[X | \cG] = (\omega \mapsto \bE[X])
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\]
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is a constant random variable.
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\end{remark}
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\paragraph{Plan}
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We will give two different proves of \autoref{conditionalexpectation}.
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The first one will use orthogonal projections.
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The second will use the Radon-Nikodym theorem.
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We'll first do the easy proof, derive some properties
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and then do the harder proof.
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\begin{lemma}
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\label{orthproj}
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Suppose $H$ is a \vocab{Hilbert space},
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i.e.~$H$ is a vector space with an inner product $\langle \cdot, \cdot \rangle_H$ which defines a norm by $\|x\|_H^2 = \langle x, x\rangle_H$
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making $H$ a complete metric space.
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For any $x \in H$ and $K \subseteq H$ closed,
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there exists a unique $z \in K$ such that the following equivalent conditions hold:
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\begin{enumerate}[(a)]
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\item $\forall y \in K : \langle x-z, y\rangle_H = 0$,
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\item $\forall y \in K: \|z-x\|_H \le \|z-x\|_H$.
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\end{enumerate}
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\end{lemma}
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\begin{proof}
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\todo{Notes}
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\end{proof}
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\begin{refproof}{conditionalexpectation}
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Almost sure uniqueness of $Z$:
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Suppose $X \in L^1$ and $Z$ and $Z'$ satisfy (a) and (b).
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We need to show that $\bP[Z \neq Z'] = 0$.
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By (a), we have $Z, Z' \in L^1(\Omega, \cG, \bP)$.
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By (b), $\bE[(Z - Z') \One_G] = 0$ for all $G \in \cG$.
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Assume that $\bP[Z > Z'] > 0$.
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Since $\{Z > Z' + \frac{1}{n}\} \uparrow \{Z > Z'\}$,
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we see that $\bP[Z > Z' + \frac{1}{n}] > 0$ for some $n$.
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However $\{Z > Z' + \frac{1}{n}\} \in \cG$,
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which is a contradiction, since
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\[
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\bE[(Z - Z') \One_{Z - Z' > \frac{1}{n}}] \ge \frac{1}{n} \bP[ Z - Z' > \frac{1}{n}] > 0.
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\]
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\bigskip
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Existence of $\bE(X | \cG)$ for $X \in L^2$:
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Let $H = L^2(\Omega, \cF, \bP)$
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and $K = L^2(\Omega, \cG, \bP)$.
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$K$ is closed, since a pointwise limit of $\cG$-measurable
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functions is $\cG$ measurable (if it exists).
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By \autoref{orthproj},
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there exists $z \in K$ such that
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\[\bE[(X - Z)^2] = \inf \{ \bE[(X- W)^2] ~|~ W \in L^2(\cG)\}\]
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and
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\begin{equation}
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\forall Y \in L^2(\cG) : \langle X - Z, Y\rangle = 0.
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\label{lec13_boxcond}
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\end{equation}
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Now, if $G \in \cG$, then $Y \coloneqq \One_G \in L^2(\cG)$
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and by \eqref{lec13_boxcond} $\bE[Z \One_G] = \bE[X \One_G]$.
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\bigskip
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Existence of $\bE(X | \cG)$ for $X \in L^1$ :
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Let $X = X^+ - X^-$.
|
||||
It suffices to show (a) and (b) for $X^+$.
|
||||
Choose bounded random variables $X_n \ge 0$ such that $X_n \uparrow X$.
|
||||
Since each $X_n \in L^2$, we can choose a version $Z_n$ of $\bE(X_n | \cG)$.
|
||||
|
||||
\begin{claim}
|
||||
$0 \overset{\text{a.s.}}{\le} Z_n \uparrow$.
|
||||
\end{claim}
|
||||
\begin{subproof}
|
||||
\todo{Notes}
|
||||
\end{subproof}
|
||||
|
||||
Define $Z(\omega) \coloneqq \limsup_{n \to \infty} Z_n(\omega)$.
|
||||
Then $Z$ is $\cG$-measurable and since $Z_n \uparrow Z$,
|
||||
by MCT, $\bE(Z \One_G) = \bE(X \One_G)$ for all $G \in \cG$.
|
||||
\end{refproof}
|
||||
|
|
255
inputs/lecture_15.tex
Normal file
255
inputs/lecture_15.tex
Normal file
|
@ -0,0 +1,255 @@
|
|||
\lecture{15}{2023-06-06}{}
|
||||
|
||||
We want to derive some properties of conditional expectation.
|
||||
|
||||
\begin{theorem}[Law of total expectation] % Thm 1
|
||||
\label{ceprop1}
|
||||
\label{totalexpectation}
|
||||
\[
|
||||
\bE[\bE[X | \cG ]] = \bE[X].
|
||||
\]
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
Apply (b) from the definition for $G = \Omega \in \cG$.
|
||||
\end{proof}
|
||||
\begin{theorem} % Thm 2
|
||||
\label{ceprop2}
|
||||
If $X$ is $\cG$-measurable, then $X = \bE[X | \cG]$ a.s..
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
Suppose $\bP[X \neq Y] > 0$.
|
||||
Without loss of generality $\bP[X > Y] > 0$.
|
||||
Hence $\bP[ X > Y + \frac{1}{n}]> 0$ for some $n \in \N$.
|
||||
Let $A \coloneqq \{X > Y + \frac{1}{n}\}$.
|
||||
% TODO
|
||||
\end{proof}
|
||||
|
||||
\begin{example}
|
||||
Suppose $X \in L^1(\bP)$, $\cG \coloneqq \sigma(X)$.
|
||||
Then $X$ is measurable with respect to $\cG$.
|
||||
Hence $\bE[X | \cG] = X$.
|
||||
\end{example}
|
||||
|
||||
\begin{theorem}[Linearity]
|
||||
\label{ceprop3}
|
||||
\label{celinearity}
|
||||
For all $a,b \in \R$
|
||||
we have
|
||||
\[
|
||||
\bE[a X_1 + bX_2 | \cG] = a \bE[X_1 | \cG] + b \bE[X_2|\cG].
|
||||
\]
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
Trivial % TODO
|
||||
\end{proof}
|
||||
|
||||
\begin{theorem}[Positivity]
|
||||
\label{ceprop4}
|
||||
% 4
|
||||
\label{cpositivity}
|
||||
If $X \ge 0$, then $\bE[X | \cG] \ge 0$ a.s.
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
Let $W $ be a version of $\E[X | \cG]$.
|
||||
Suppose $\bP[ W < 0] > 0$.
|
||||
Then $G \coloneqq \{W < -\frac{1}{n}\} \in \cG$
|
||||
For some $n \in \N$, we have $\bP[G] > 0$.
|
||||
However it follows that
|
||||
\[
|
||||
\int_G \bP[X | \cG] \dif \bP \le -\frac{1}{n} \bP[G] < 0 \le \int_G X \dif \bP.
|
||||
\]
|
||||
\end{proof}
|
||||
\begin{theorem}[Conditional monotone convergence theorem]
|
||||
\label{ceprop5}
|
||||
% 5
|
||||
\label{mcmt}
|
||||
Let $X_n,X \in L^1(\Omega, \cF, \bP)$.
|
||||
Suppose $X_n \ge 0$ with $X_n \uparrow X$.
|
||||
Then $\bE[X_n|\cG] \uparrow \bE[X|\cG]$.
|
||||
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
Let $Z_n$ be a version of $\bE[X_n | Y]$.
|
||||
Since $X_n \ge 0$ and $X_n \uparrow$,
|
||||
by \autoref{cpositivity},
|
||||
we have
|
||||
\[
|
||||
\bE[X_n | \cG] \overset{\text{a.s.}}{\ge } 0
|
||||
\]
|
||||
and
|
||||
\[
|
||||
\bE[X_n | \cG] \uparrow \text{a.s.}
|
||||
\]
|
||||
(consider $X_{n+1} - X_n$ ).
|
||||
|
||||
Define $Z \coloneqq \limsup_{n \to \infty} Z_n$.
|
||||
Then $Z$ is $\cG$-measurable
|
||||
and $Z_n \uparrow Z$ a.s.
|
||||
|
||||
Take some $G \in \cG$.
|
||||
We know by (b) % TODO REF
|
||||
that $\be[Z_n \One_G] = \bE[X_n \One_G]$.
|
||||
The LHS increases to $\bE[Z \One_G]$ by the monotone
|
||||
convergence theorem.
|
||||
Again by MCT, $\bE[X_n \One_G]$ increases to
|
||||
$\bE[X \One_G]$.
|
||||
Hence $Z$ is a version of $\bE[X | \cG]$.
|
||||
\end{proof}
|
||||
|
||||
\begin{theorem}[Conditional Fatou]
|
||||
\label{ceprop6}
|
||||
\label{cfatou}
|
||||
Let $X_n \in L^1(\Omega, \cF, \bP)$, $X_n \ge 0$.
|
||||
Then
|
||||
\[
|
||||
\bE[ \liminf_{n \to \infty} X_n | \cG] \le \liminf_{n \to \infty} \bE[X_n | \cG].
|
||||
\]
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
\todo{in the notes}
|
||||
\end{proof}
|
||||
\begin{theorem}[Conditional dominated convergence theorem]
|
||||
\label{ceprop7}
|
||||
\label{cdct}
|
||||
Let $X_n,X \in L^1(\Omega, \cF, \bP)$.
|
||||
Suppose $|X_n(\omega)| < X(\omega)$ a.e.~
|
||||
and $\int |X| \dif \bP < \infty$.
|
||||
Then $X_n(\omega) \to X\left( \omega \right) \implies \bE[ X_n | \cG] \to \bE[X | \cG]$.
|
||||
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
\todo{in the notes}
|
||||
\end{proof}
|
||||
|
||||
Recall
|
||||
\begin{theorem}[Jensen's inequality]
|
||||
If $c : \R \to \R$ is convex and $\bE[|c \circ X|] < \infty$,
|
||||
then $\bE[c \circ X] \ge c(\bE[X])$.
|
||||
\end{theorem}
|
||||
|
||||
For conditional expectation, we have
|
||||
\begin{theorem}[Conditional Jensen's inequality]
|
||||
\label{ceprop8}
|
||||
\label{cjensen}
|
||||
Let $X \in L^1(\Omega, \cF, \bP)$.
|
||||
If $c : \R \to \R$ is convex and $\bE[|c \circ X|] < \infty$,
|
||||
then $\bE[c \circ X | \cG] \ge c(\bE[X | \cG])$ a.s.
|
||||
\end{theorem}
|
||||
\begin{fact}
|
||||
\label{convapprox}
|
||||
If $c$ is convex, then there are two sequences of real numbers
|
||||
$a_n, b_n \in \R$
|
||||
such that
|
||||
\[
|
||||
c(x) = \sup_n(a_n x + b_n).
|
||||
\]
|
||||
\end{fact}
|
||||
\begin{refproof}{cjensen}
|
||||
By \autoref{convapprox}, $c(x) \ge a_n X + b_n$
|
||||
for all $n$.
|
||||
Hence
|
||||
\[
|
||||
\bE[c(X) | \cG] \ge a_n \bE[X | \cG] + \bE[b_n | \cG]
|
||||
= a_n \bE[X | \cG] + b_n \text{a.s.}
|
||||
\]
|
||||
for all $n$.
|
||||
Using that a countable union of sets o f measure zero has measure zero,
|
||||
we conclude that a.s~this happens simultaneously for all $n$.
|
||||
Hence
|
||||
\[
|
||||
\bE[c(X) | \cG] \ge \sup_n (a_n \bE[X | \cG] + b_n) \overset{\text{\autoref{convapprox}}}{=} c(\bE(X | \cG)).
|
||||
\]
|
||||
\end{refproof}
|
||||
|
||||
Recall
|
||||
\begin{theorem}[Hölder's inequality]
|
||||
Let $p,q \ge 1$ such that $\frac{1}{p} + \frac{1}{q} = 1$.
|
||||
Suppose $X \in L^p(\bP)$ and $Y \in L^q(\bP)$.
|
||||
Then
|
||||
\[
|
||||
\bE(X Y) \le \underbrace{\bE(|X|^p)^{\frac{1}{p}}}_{\text{\reflectbox{$\coloneqq$}} \|X\|_{L^p}} \bE(|Y|^q)^{\frac{1}{q}}.
|
||||
\]
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Conditional Hölder's inequality]
|
||||
\label{ceprop9}
|
||||
\label{choelder}
|
||||
Let $p,q \ge 1$ such that $\frac{1}{p} + \frac{1}{q} = 1$.
|
||||
Suppose $X \in L^p(\bP)$ and $Y \in L^q(\bP)$.
|
||||
Then
|
||||
\[
|
||||
\bE(X Y | \cG) \le \bE(|X|^p | \cG)^{\frac{1}{p}} \bE(|Y|^q | \cG)^{\frac{1}{q}}.
|
||||
\]
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
Similar to the proof of Hölder's inequality.
|
||||
\todo{Exercise}
|
||||
\end{proof}
|
||||
|
||||
\begin{theorem}[Tower property]
|
||||
% 10
|
||||
\label{ceprop10}
|
||||
\label{ctower}
|
||||
Suppose $\cF \supset \cG \supset \cH$ are sub-$\sigma$-algebras.
|
||||
Then
|
||||
\[
|
||||
\bE\left[\bE[X | \cG] \mid \cH\right] = \bE[X | \cH].
|
||||
\]
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
\todo{Exercise}
|
||||
\end{proof}
|
||||
|
||||
\begin{theorem}[Taking out what is known]
|
||||
% 11
|
||||
\label{ceprop11}
|
||||
\label{takingoutwhatisknown}
|
||||
|
||||
If $Y$ is $\cG$-measurable and bounded, then
|
||||
\[
|
||||
\bE[YX| \cG] \overset{\text{a.s.}}{=} Y \bE[X | \cG].
|
||||
\]
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
Assume w.l.o.g.~$X \ge 0$.
|
||||
Assume $Y = \One_B$, then $Y$ simple, then take the limit (using that $Y$ is bounded).
|
||||
\todo{Exercise}
|
||||
\end{proof}
|
||||
|
||||
\begin{definition}
|
||||
Let $\cG$ and $\cH$ be $\sigma$-algebras.
|
||||
We call $\cG$ and $\cH$ \vocab[$\sigma$-algebra!independent]{independent},
|
||||
if % TODO
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Role of independence]
|
||||
\label{ceprop12}
|
||||
\label{roleofindependence}
|
||||
If $\cH$ is a sub-$\sigma$-algebra of $\cF$ and $\cH$ is independent
|
||||
of $\sigma(\sigma(X), \cG)$, then
|
||||
\[
|
||||
\bE[X | \sigma(\cG, \cH)] \overset{\text{a.s.}}{=} \bE[X | \cG].
|
||||
\]
|
||||
\end{theorem}
|
||||
\begin{example}
|
||||
If $X$ is independent of $\cG$,
|
||||
then $\bE[X | \cG] \overset{\text{a.s.}}{=} \bE[X]$.
|
||||
\end{example}
|
||||
\begin{example}[Martingale property of the simple random walk]
|
||||
Suppose $X_1,X_2,\ldots$ are i.i.d.~with $\bP[X_i = 1] = \bP[X_i = -1] = \frac{1}{2}$.
|
||||
Let $S_n \coloneqq \sum_{i=1}^n X_i$ be the \vocab{simple random walk}.
|
||||
Let $\cF$ denote the $\sigma$-algebra on the product space.
|
||||
Define $\cF_n \coloneqq \sigma(X_1,\ldots)$.
|
||||
Intuitively, $\cF_n$ contains all the information gathered until time $n$.
|
||||
We have $\cF_1 \subset \cF_2 \subset \cF_3 \subset \ldots$
|
||||
|
||||
For $\bE[S_{n+1} | \cF_n]$ we obtain
|
||||
\begin{IEEEeqnarray*}{rCl}
|
||||
\bE[S_{n+1} | \cF_n] &\overset{\autoref{celinearity}}{=}&
|
||||
\bE[S_n | \cF_n] + \bE[X_{n+1} | \cF_n]\\
|
||||
&\overset{\text{a.s.}}{=}& S_n + \bE[X_{n+1} | \cF_n]\\
|
||||
&\overset{\text{\autoref{ceprop12}}}{=}& S_{n} + \bE[X_n]\\
|
||||
&=& S_n
|
||||
\end{IEEEeqnarray*}
|
||||
|
||||
\end{example}
|
|
@ -1,3 +1,4 @@
|
|||
\lecture{2}{}{}
|
||||
\section{Independence and product measures}
|
||||
|
||||
In order to define the notion of independence, we first need to construct
|
||||
|
|
|
@ -1,3 +1,4 @@
|
|||
\lecture{3}{}{}
|
||||
\todo{Lecture 3 needs to be finished}
|
||||
\begin{notation}
|
||||
Let $\cB_n$ denote $\cB(\R^n)$.
|
||||
|
|
|
@ -1,4 +1,4 @@
|
|||
% Lecture 5 2023-04-21
|
||||
\lecture{5}{2023-04-21}{}
|
||||
\subsection{The laws of large numbers}
|
||||
|
||||
|
||||
|
|
|
@ -103,3 +103,4 @@
|
|||
\DeclareSimpleMathOperator{Exp}
|
||||
|
||||
\newcommand*\dif{\mathop{}\!\mathrm{d}}
|
||||
\newcommand\lecture[3]{{\color{gray}\hfill Lecture #1 (#2)}}
|
||||
|
|
Loading…
Reference in a new issue