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\begin{document}
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\begin{center} \bf
Statistics 530: Probability Theory \\
Homework No. 8
\end{center}
\medskip
\noindent
{\sc Reading}\
\smallskip
Press on with your reading of Durrett through page 158.
\smallskip
\begin{Problem} {\sc Quick Shots}. For $\lambda >0$, justify the following assertions:
\begin{itemize}
\item $\phi_1(t)=\exp(\lambda(e^{it}-1))$ is a characteristic function.
\item $\phi_2(t)=\exp(\lambda(1-e^{it}))$ is NOT a characteristic function.
\item $\phi_3(t)=\exp(\lambda(\cos t -1))$ is a characteristic function.
\end{itemize}
\end{Problem}
\begin{Problem} Suppose that $x_n=(x_{n,1}, x_{n,2}, \dots , x_{n,d})$ is a sequence of vectors in $\R^d$. Suppose that $x_{n,j}\geq 0$
for all $n\geq 1$ and $1\leq j \leq d$, and
suppose that there is an $A$ such that for $n \rightarrow \infty$,
\begin{align*}
x_{n,1}+ x_{n,2}+ \dots + x_{n,d} & \rightarrow d A \\
x_{n,1}^2+ x_{n,2}^2+ \dots + x_{n,d}^2 & \rightarrow d A^2
\end{align*}
Show that $(x_{n,1}, x_{n,2}, \dots , x_{n,d}) \rightarrow (A, A, ...,A)$ as $n \rightarrow \infty$.
\smallskip
\noindent{Hint:} This is a cool real variable fact, but why would I pester you with it at this point in your life? It is because this elementary fact
illustrates the principle used to prove Levy's continuity theorem. For the \emph{determining condition} consider the possibility that we had
equality in these two relations (not just limits). Show how the case of equality in Cauchy-Schwarz (or the case of equality in Jensen's
inequality) would then imply that $(x_{n,1}, x_{n,2}, \dots , x_{n,d})$ is a constant vector. Now use compactness and subsequence arguments to get the conjectured
limit theorem.
\end{Problem}
\begin{Problem} Suppose that $f_n(x)$ is a sequence of densities such that $f_n(x)$ converges to $f(x)$ for all $x \in \R$. Show that if $f$
is \emph{also} a density, then
$$
\lim_{n \rightarrow \infty} \int_{-\infty}^\infty |f_n(x) -f(x) | \, dx \rightarrow 0 \quad \text{as } n \rightarrow \infty.
$$
{\sc Hint:} Even more than usual, you have to use all of the hypotheses! Bigger hint --- What can you say about the sequence of functions
$$
g_n(x) = \{f(x) -f_n(x)\} 1(x: f(x) -f_n(x)\geq 0\}
$$
{\sc Note:} This is a problem we could have discussed in the first couple of weeks. It is left until now because earlier
we ourselves did not give ourselves licence to think about densities. It is a good
discipline to avoid them whenever it is sensible to do so.
\end{Problem}
\begin{Problem} Complete the outline below to show that if for each $k=0,1,2,...$ the sequence of random variables $Z_n$ satisfies
$$
E(Z_n^k) \rightarrow \frac{1}{1+k} \quad \text{as } n \rightarrow \infty
$$
then $Z_n$ converges in distribution to the uniform distribution on $[0,1]$ as $n \rightarrow \infty$. Here is the outline:
\begin{itemize}
\item Show that the sequence of distributions of the $Z_n$ is tight.
\item Show that if $E(W^k)=\frac{1}{1+k}$ for all $k=0,1,...$ then $W$ is uniformly distributed on $[0,1]$. Please confine your tools to results that we have proved in class. No magic
incantations, SVP.
\item Follow the pattern we used to prove Levy's continuity theorem to complete the solution of the problem. You will probably want to use a subsequence argument and perhaps an argument by
contradiction.
\end{itemize}
{\sc Note:} By understanding this exercise and related exercises, you will get to understand the pattern behind Levy's continuity theorem. This is an admission ticket to a reasonably small club, much
smaller than the Masons, for example.
\end{Problem}
\begin{Problem} Suppose that $X$ is a random variable with $P(X >0)=1$ and with a finite mean $\mu>0$. Complete the following plan:
\begin{itemize}
\item Explain why $f(x)=\mu^{-1} P(X > x)$ is a probability density on $[0,\infty)$.
\item Suppose that $Y$ is a random variable with density $f$. Calculate the characteristic function $\psi(t)$ of $Y$ in terms of the characteristic function of $\phi(t)$ of $X$. Note: The answer
is a simple algebraic function of $\mu$, $\phi(t)$, and constants.
\item Check your formula by confirming directly from the formula that $\psi(0)=1$.
\item Suppose $X$ has the exponential distribution with mean one. What is the distribution of $Y$?
\end{itemize}
{\sc Note:} This is a very general recipe and --- more often than one might suspect ---
the examination of $Y$ will tell us something interesting about the distribution of $X$. Here for example, the
uniform continuity of $\psi(t)$ gives us some additional ``continuity information" about $\phi(t)$ in a form that one could never conjectured without this auxiliary calculation.
It is very handy to have ``standard recipes" that tell us how to morph one problem into another.
You should keep a list of these for future uses. Sometimes such morphings just send us around in
circles, but at luckier times we end up discovering something amusing --- or even genuinely new.
\end{Problem}
\end{document}