moved role of independence
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4 changed files with 25 additions and 39 deletions
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@ -219,22 +219,28 @@ Assume $Y = \One_B$, then $Y$ simple, then take the limit (using that $Y$ is bou
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\begin{definition}
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\begin{definition}
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Let $\cG$ and $\cH$ be $\sigma$-algebras.
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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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We call $\cG$ and $\cH$ \vocab[$\sigma$-algebra!independent]{independent},
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if % TODO
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\todo{TODO}
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\end{definition}
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\end{definition}
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\begin{theorem}[Role of independence]
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\begin{theorem}[Role of independence]
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\label{ceprop12}
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\label{ceprop12}
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\label{roleofindependence}
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\label{ceroleofindependence}
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If $\cH$ is a sub-$\sigma$-algebra of $\cF$ and $\cH$ is independent
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Let $X$ be a random variable,
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of $\sigma(\sigma(X), \cG)$, then
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and let $\cG, \cH$ be $\sigma$-algebras.
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\[
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\bE[X | \sigma(\cG, \cH)] \overset{\text{a.s.}}{=} \bE[X | \cG].
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If $\cH$ is independent of $\sigma\left( \sigma(X), \cG \right)$,
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\]
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then
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\[
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\bE[X | \sigma(\cG, \cH)] \overset{\text{a.s.}}{=} \bE[X | \cG].
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\]
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In particular, if $X$ is independent of $\cG$,
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then
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\[
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\bE[X | \cG] \overset{\text{a.s.}}{=} \bE[X].
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\]
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\end{theorem}
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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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\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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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 $S_n \coloneqq \sum_{i=1}^n X_i$ be the \vocab{simple random walk}.
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@ -1,29 +1,8 @@
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\lecture{16}{2023-06-13}{}
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\lecture{16}{2023-06-13}{}
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\subsection{Conditional expectation}
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% \subsection{Conditional expectation}
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\begin{theorem}
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\begin{refproof}{ceroleofindependence}
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\label{ceprop11}
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\label{ceroleofindependence}
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Let $X$ be a random variable,
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and let $\cG, \cH$ be $\sigma$-algebras.
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If $\cH$ is independent of $\sigma\left( \sigma(X), \cG \right)$,
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then
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\[
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\bE[X | \sigma(\cG, \cH)] \overset{\text{a.s.}}{=} \bE[X | \cG].
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\]
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In particular, if $X$ is independent of $\cG$,
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then
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\[
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\bE[X | \cG] \overset{\text{a.s.}}{=} \bE[X].
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\]
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\end{theorem}
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\todo{Definition of independence wrt a $\sigma$-algebra}
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\begin{proof}
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Let $\cH$ be independent of $\sigma(\sigma(X), \cG)$.
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Let $\cH$ be independent of $\sigma(\sigma(X), \cG)$.
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Then for all $H \in \cH$, we have that $\One_H$
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Then for all $H \in \cH$, we have that $\One_H$
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and any random variable measurable with respect to either $\sigma(X)$
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and any random variable measurable with respect to either $\sigma(X)$
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@ -50,7 +29,7 @@
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The claim of the theorem follows by the uniqueness of conditional expectation.
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The claim of the theorem follows by the uniqueness of conditional expectation.
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To deduce the second statement, choose $\cG = \{\emptyset, \Omega\}$.
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To deduce the second statement, choose $\cG = \{\emptyset, \Omega\}$.
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\end{proof}
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\end{refproof}
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\subsection{The Radon Nikodym theorem}
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\subsection{The Radon Nikodym theorem}
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@ -1,7 +1,9 @@
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\lecture{17}{2023-06-15}{}
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\lecture{17}{2023-06-15}{}
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\begin{definition}[Stochastic process]
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\begin{definition}[Stochastic process]
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% TODO
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A \vocab{stochastic process} is a collection of random
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variables $(X_t)_{t \in T}$ for some index set $T$.
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In this lecture we will consider the case $T = \N$.
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\end{definition}
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\end{definition}
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\begin{goal}
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\begin{goal}
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@ -61,7 +61,7 @@ However, some subsets can be easily described, e.g.
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\begin{fact}\label{lec19f2}
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\begin{fact}\label{lec19f2}
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If $(X_n)_n$ is uniformly integrable,
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If $(X_n)_n$ is uniformly integrable,
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then $(X_n)_n$ is bounded in $L^1$.k:w
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then $(X_n)_n$ is bounded in $L^1$.
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\end{fact}
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\end{fact}
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\begin{fact}\label{lec19f3}
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\begin{fact}\label{lec19f3}
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@ -223,7 +223,7 @@ Let $(\Omega, \cF, \bP)$ as always and let $(\cF_n)_n$ always be a filtration.
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to $X$ in $L^p$.
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to $X$ in $L^p$.
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\end{theorem}
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\end{theorem}
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\begin{proof}
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\begin{proof}
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\todo{TODO}
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\end{proof}
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\end{proof}
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\begin{theorem}
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\begin{theorem}
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@ -231,7 +231,6 @@ Let $(\Omega, \cF, \bP)$ as always and let $(\cF_n)_n$ always be a filtration.
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Let $(X_n)_n$ be a martingale bounded in $L^p$.
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Let $(X_n)_n$ be a martingale bounded in $L^p$.
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Then there exists a random variable $X \in L^p$, such that
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Then there exists a random variable $X \in L^p$, such that
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$X_n = \bE[X | \cF_n]$ for all $n$.
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$X_n = \bE[X | \cF_n]$ for all $n$.
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\end{theorem}
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\end{theorem}
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