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\begin{center} \bf
Statistics 930: Probability Theory \\
Homework No. 11
\end{center}
{\sc General Comments:} Some of these are intuitive, foundational problems where it is easy to just make a bunch of assertions that
are reasonable and that end with the desired conclusion. To really ring the bell, one has to be brutally
attentive to the logic, and, if you say the word
``obvious," then you have almost surely made a mistake. In this context, the best proofs tend to have the fewest words and the tightest logic.
\begin{Problem} Suppose $\{\FF_n\}$ is an increasing sequence of $\sigma$-fields, and set
$$
\PP=\cup \FF_n \quad \text{and} \quad \FF= \sigma \{\cup \FF_n\}=\sigma\{\PP\}.
$$
Given a probability measure $P$ on $\FF$, we then consider the set
$$
\GG=\{A \in \FF: \forall \epsilon>0, \, \exists A_\epsilon \in \PP, \, \text{such that } P (A \Delta A_\epsilon) \leq \epsilon \}.
$$
\smallskip
Show that $\GG$ is a $\lambda$-system and use this to deduce that $\FF \subset \GG$. In longhand, this says that any
element of $\FF$ can be approximated
as well as we like by an element of $\PP$.
\end{Problem}
\begin{Problem} Suppose that $\{X_1, X_2, \ldots\}$ is an infinite sequence of (possibly dependent)
random variables and let $\FF$ be the smallest sigma-field
for which these are all measurable. Suppose $X$ is a bounded, $\FF$-measurable random variable. Show that there for each $\epsilon >0$,
there is an $n$ and a function $f: \R^n \rightarrow \R$ such that
$$
\E [| X-f(X_1,X_2, \ldots, X_n)|] \leq \epsilon.
$$
Remark. You probably want to make use of Problem 1.
\end{Problem}
\begin{Problem} Consider sequences of random variables such that
$X_n$ converges in probability to $1$ as $n \rightarrow \infty$ and $\E[Y_n]=\alpha$ for all $n=1,...$.
\begin{itemize}
\item Give an example such that the product is $X_nY_n$ is integrable for each $n$, but
the sequence $\E[X_nY_n]$ does not converge to $\alpha$ as $n\rightarrow \infty$.
\item Now assume that the collection
$\{X_n, Y_n, X_nY_n: n=1,2, \ldots\}$ is uniformly integrable, and
show that $\E[X_nY_n]$ converges to $\alpha$ as $n \rightarrow \infty$.
Examine your proof to see if you can weaken the uniform integrability condition.
\end{itemize}
\end{Problem}
\begin{Problem}
\begin{itemize}
\item Let $C$ be the collection of
functions $f: \Omega \rightarrow [0,1]$. This a convex set, and its easy to see that the extreme points
are those functions that take values in the two-point set $\{0,1\}$, but this not too useful in probability theory.
State and prove a (slightly) better version for a probability space with $\Omega=[0,1]$.
\item Let $C$ be the set of probability measures on the Borel subsets of $[0,1]$. For each $x$ let $\mu_x$ be the probability measure
such that $\mu_x(\{x\})=1$. Show that for each $x$ the measure $\mu_x$ is an extreme point of $C$ and show
that every extreme point of $C$ can be written as $\mu_x$ for some $x$. If you can't prove the general result, prove the best result you can.
\item Let $C$ be the set of twice differentiable, convex functions $f:[0,1] \rightarrow [0,1]$ such that $f(0)=1$ and $f(1)=0$. Show that $C$ is convex,
and show that $f(x)=1-x$ is the only extreme point of $C$. This gives us a pretty big set with just one extreme point, but this may not be too surprising
since we saw in class that the unit ball in $L^1[0,1]$ has no extreme points.
\end{itemize}
\end{Problem}
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