diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex index a3b53b8..4ba12b1 100644 --- a/inputs/lecture_15.tex +++ b/inputs/lecture_15.tex @@ -219,22 +219,28 @@ Assume $Y = \One_B$, then $Y$ simple, then take the limit (using that $Y$ is bou \begin{definition} Let $\cG$ and $\cH$ be $\sigma$-algebras. We call $\cG$ and $\cH$ \vocab[$\sigma$-algebra!independent]{independent}, - if % TODO + \todo{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 - \[ - \bE[X | \sigma(\cG, \cH)] \overset{\text{a.s.}}{=} \bE[X | \cG]. - \] + \label{ceroleofindependence} + Let $X$ be a random variable, + and let $\cG, \cH$ be $\sigma$-algebras. + + If $\cH$ is independent of $\sigma\left( \sigma(X), \cG \right)$, + then + \[ + \bE[X | \sigma(\cG, \cH)] \overset{\text{a.s.}}{=} \bE[X | \cG]. + \] + + In particular, if $X$ is independent of $\cG$, + then + \[ + \bE[X | \cG] \overset{\text{a.s.}}{=} \bE[X]. + \] \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}. diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index e78e929..91182b0 100644 --- a/inputs/lecture_16.tex +++ b/inputs/lecture_16.tex @@ -1,29 +1,8 @@ \lecture{16}{2023-06-13}{} -\subsection{Conditional expectation} +% \subsection{Conditional expectation} -\begin{theorem} - \label{ceprop11} - \label{ceroleofindependence} - Let $X$ be a random variable, - and let $\cG, \cH$ be $\sigma$-algebras. - - If $\cH$ is independent of $\sigma\left( \sigma(X), \cG \right)$, - then - \[ - \bE[X | \sigma(\cG, \cH)] \overset{\text{a.s.}}{=} \bE[X | \cG]. - \] - - In particular, if $X$ is independent of $\cG$, - then - \[ - \bE[X | \cG] \overset{\text{a.s.}}{=} \bE[X]. - \] -\end{theorem} - -\todo{Definition of independence wrt a $\sigma$-algebra} - -\begin{proof} +\begin{refproof}{ceroleofindependence} Let $\cH$ be independent of $\sigma(\sigma(X), \cG)$. Then for all $H \in \cH$, we have that $\One_H$ and any random variable measurable with respect to either $\sigma(X)$ @@ -50,7 +29,7 @@ The claim of the theorem follows by the uniqueness of conditional expectation. To deduce the second statement, choose $\cG = \{\emptyset, \Omega\}$. -\end{proof} +\end{refproof} \subsection{The Radon Nikodym theorem} diff --git a/inputs/lecture_17.tex b/inputs/lecture_17.tex index 164472e..ae63e06 100644 --- a/inputs/lecture_17.tex +++ b/inputs/lecture_17.tex @@ -1,7 +1,9 @@ \lecture{17}{2023-06-15}{} \begin{definition}[Stochastic process] - % TODO + A \vocab{stochastic process} is a collection of random + variables $(X_t)_{t \in T}$ for some index set $T$. + In this lecture we will consider the case $T = \N$. \end{definition} \begin{goal} diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index 6766371..505d65e 100644 --- a/inputs/lecture_19.tex +++ b/inputs/lecture_19.tex @@ -61,7 +61,7 @@ However, some subsets can be easily described, e.g. \begin{fact}\label{lec19f2} If $(X_n)_n$ is uniformly integrable, - then $(X_n)_n$ is bounded in $L^1$.k:w + then $(X_n)_n$ is bounded in $L^1$. \end{fact} \begin{fact}\label{lec19f3} @@ -223,7 +223,7 @@ Let $(\Omega, \cF, \bP)$ as always and let $(\cF_n)_n$ always be a filtration. to $X$ in $L^p$. \end{theorem} \begin{proof} - + \todo{TODO} \end{proof} \begin{theorem} @@ -231,7 +231,6 @@ Let $(\Omega, \cF, \bP)$ as always and let $(\cF_n)_n$ always be a filtration. Let $(X_n)_n$ be a martingale bounded in $L^p$. Then there exists a random variable $X \in L^p$, such that $X_n = \bE[X | \cF_n]$ for all $n$. - \end{theorem}