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\begin{document}
\begin{center}
{\bf Probability 1\\
CEU Budapest, fall semester 2014}\\
Imre Péter Tóth\\
{\bf Homework sheet 4 -- due on 30.10.2014 -- and exercises for practice}\\
\end{center}


\begin{enumerate}[4.1]
%
\item \textbf{(homework)} Durrett~\cite{durrett}, Exercise 3.2.11
%
\item \textbf{(homework)} \emph{Poisson approximation of the binomial distribution.}
Fix $0<\lambda\in\IR$. Show that if $X_n$ has binomial distribution with parameters $(n,p)$ such that $np\to\lambda$ as $n\to\infty$, then $X_n$ converges to $Poi(\lambda)$ weakly.
This can be done in a completely elementary way, using your favourite definition of weak convergence, or by using one of the stronger tools of weak convergence.
%
% \item \textbf{(homework)} Let $X$ be uniformly distributed on $[-1;1]$, and set $Y_n=nX$.
% \begin{enumerate}[a.)]
%  \item Calculate the characteristic function $\psi_n$ of $Y_n$.
%  \item Calculate the pointwise limit $\li \psi_n(t)$, if it exists.
%  \item Does (the distribution of) $Y_n$ have a weak limit?
%  \item How come?
% \end{enumerate}
%
\item Let $X_1,X_2,\dots$ be independent random variables such that
$$
\prob(X_n=n^2-1)=\frac{1}{n^2},\quad \prob(X_n=-1)=1-\frac{1}{n^2}.
$$
Show that $\expect X_n=0$ for every $n$, but
$$
\li \frac{X_1+\dots X_n}{n}=-1
$$
almost surely.
%
\item \textbf{(homework)} Let $F:\IR\to[0,1]$ be a probability distribution function, and let $Y$ be a random variable which is uniformly distributed in $[0,1]$. Let
$X=\sup\{x | F(x)<Y\}$. Show that the distribution function of $X$ is exactly $F$.
%
\item Let $X_1,X_2,\dots,X_n$ be i.i.d. random variables with $\expect X_1=0$ and $\expect (X_1^4)<\infty$ and set $S_n=X_1+\cdots+X_n$. Show that there is a $C<\infty$ such that
$\expect(S_n^4)\le C n^2$.
%
\item 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 3.2.6
%
\item Durrett~\cite{durrett}, Exercise 3.2.9
%
\item \textbf{(homework)} Durrett~\cite{durrett}, Exercise 3.2.12
%
\item Durrett~\cite{durrett}, Exercise 3.2.14
%
\item Durrett~\cite{durrett}, Exercise 3.2.15
%
\end{enumerate}


\begin{thebibliography}{1}
 \bibitem{durrett}Durrett, R. \emph{Probability: Theory and Examples, 4th edition}. Cambridge University Press (2010)
\end{thebibliography}


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