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\begin{document}
\begin{center}
{\bf Probability 1\\
CEU Budapest, fall semester 2013}\\
Imre Péter Tóth\\
{\bf Homework sheet 4 -- due on 14.10.2013 -- and exercises for practice}\\
\end{center}


\begin{enumerate}[4.1]
%
\item \textbf{(homework)} For real numbers $a_1,a_2,a_3,\dots$ define the infinite product $\prod\limits_{k=1}^{\infty}a_k$ as
$$
\prod_{k=1}^{\infty}a_k:=\lim_{n\to\infty}\prod_{k=1}^{n}a_k,
$$
whenever this limit exists.

Let $p_1,p_2,p_3,\dots$ satisfy $0\le p_k<1$ for all $k$. Show that $\prod\limits_{k=1}^{\infty}(1-p_k)>0$ if and only if $\sum\limits_{k=1}^{\infty}p_k <\infty$.

\emph{(Hint: estimate the logarithm of $(1-p)$ with $p$.)}
%
\item Durrett~\cite{durrett}, Exercise 3.3.1
%
\item Durrett~\cite{durrett}, Exercise 3.3.3
%
\item Durrett~\cite{durrett}, Exercise 3.3.9
%
\item \textbf{(homework)} Durrett~\cite{durrett}, Exercise 3.3.10. Show also that independence is needed.
%
\item Durrett~\cite{durrett}, Exercise 3.3.11
%
\item \textbf{(homework)} Durrett~\cite{durrett}, Exercise 3.3.12
%
\item Durrett~\cite{durrett}, Exercise 3.3.13
%
\end{enumerate}

\begin{thebibliography}{1}
 \bibitem{durrett}Durrett, R. \emph{Probability: Theory and Examples}. Cambridge University Press (2010)
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