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\begin{document}
\begin{center}
{\bf Probability 1\\
CEU Budapest, fall semester 2013}\\
Imre Péter Tóth\\
{\bf Homework sheet 5 -- due on 28.10.2013 -- and exercises for practice}\\
\end{center}


\begin{enumerate}[5.1]
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\item \textbf{(homework)} Let $X_1,X_2,\dots$ be i.i.d. random variables with density (w.r.t. Lebesgue measure)
$f(x)=\frac{1}{\pi}\frac{1}{1+x^2}$. (So they have the Cauchy distribution.) Find the weak limit (as $n\to\infty$)
of the average
$$\frac{X_1+\dots +X_n}{n}.$$
\emph{Warning: this is not hard, but also not as trivial as it may seem. Hint: a possible solution is using characteristic functions. Calculating the characteristic function of the Cauchy distribution is a little tricky, but you can look it up.}
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\item Durrett~\cite{durrett}, Exercise 3.3.20
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\item \textbf{(homework)} Durrett~\cite{durrett}, Exercise 3.4.4
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\item Durrett~\cite{durrett}, Exercise 3.4.5
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\item \textbf{(homework)} Durrett~\cite{durrett}, Exercise 3.6.1
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\item Durrett~\cite{durrett}, Exercise 3.6.2
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\begin{thebibliography}{1}
 \bibitem{durrett}Durrett, R. \emph{Probability: Theory and Examples}. Cambridge University Press (2010)
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