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\begin{document}
\begin{center}
{\bf Probability 1\\
CEU Budapest, fall semester 2013}\\
Imre Péter Tóth\\
{\bf Homework sheet 6 -- due on 04.10.2013 -- and exercises for practice}\\
\end{center}


\begin{enumerate}[6.1]
%
\item Let $X_1,X_2,\dots,X_n$ be i.i.d. random variables. Prove that the following two statements are equivalent:
\begin{enumerate}[(i)]
 \item $\expect |X_i|<\infty$.
 \item $\prob(|X_n|>n \text{ for infinitely many $n$-s})=0$.
\end{enumerate}
%
\item \textbf{(homework)} Prove that for \emph{any} sequence $X_1,X_2,\dots$ of random variables (real valued, defined on the same probability space) there exists a sequence $c_1,c_2,\dots$ of numbers such that
$$\frac{X_n}{c_n}\to 0 \text{ almost surely}.$$
%
\item Let the random variables $X_1,X_2,\dots,X_n,\dots$ and $X$ be defined on the same probability space. Prove that the following two statements are equivalent:
\begin{enumerate}[(i)]
 \item $X_n\to X$ in probability as $n\to\infty$.
 \item From every subsequence $\{n_k\}_{k=1}^\infty$ a sub-subsequence $\{n_{k_j}\}_{j=1}^\infty$ can be chosen such that $X_{n_{k_j}}\to X$ almost surely as $j\to\infty$.
\end{enumerate}
%
\item \textbf{(homework)} Let $X_1,X_2,\dots$ be independent such that $X_n$ has $Bernoulli(p_n)$ distribution. Determine what property the sequence $p_n$ has to satisfy so that
\begin{enumerate}[(a)]
 \item $X_n\to X$ in probability as $n\to\infty$
 \item $X_n\to X$ almost surely as $n\to\infty$.
\end{enumerate}
%
\item Let $X_1,X_2,\dots$ be independent random variables. Show that $\prob(\sup_n X_n<\infty)=1$ if and only if there is some $A\in\IR$ for which $\sum_{n=1}^\infty \prob(X_n>A)<\infty$.
%
\item Let $X_1,X_2,\dots$ be independent exponentially distributed random variables such that $X_n$ has parameter $\lambda_n$. Let $S_n:=\sum_{i=1}^n X_i$. Show that if $\sum_{n=1}^\infty \frac{1}{\lambda_n}=\infty$, then $S_n\to\infty$ almost surely, but if $\sum_{n=1}^\infty \frac{1}{\lambda_n}<\infty$, then $S_n\to S$ almost surely, where $S$ is some random variable which is almost surely finite.
%
\item Let $X_1,X_2,\dots$ be i.i.d. random variables with distribution $Bernoulli(p)$ for some $p\in(0;1)$ but $p\ne\frac12$. Let $Y:=\sum_{n=1}^\infty 2^{-n} X_n$. (The sum is absolutely convergent.) Show that the distribution of $Y$ is continuous, but singular w.r.t. Lebesgue measure. 
%
\item \textbf{(homework)} Let the random variables $X_1,X_2,\dots,X_n,\dots$ and $X$ be defined on the same probability space and suppose that $X_n\to X$ in probability as $n\to\infty$.
\begin{enumerate}[(a)]
 \item If $f:\IR\to\IR$ is a continuous function, $Y_n=f(X_n)$ and $Y=f(X)$, show that $Y_n\to Y$ in probability as $n\to\infty$.
 \item Show that if the $X_n$ are almost surely uniformly bounded [that is: there exists a constant $M<\infty$ such that $\prob(\forall n\in\IN\, |X_n|\le M)=1$], then $\lim_{n\to\infty}\expect X_n =\expect X$.
 \item Show, through an example, that for the previous statement, tha condition of boundedness is needed.
\end{enumerate}
%
\item Let the random variables $X_1,X_2,\dots$, $Y_1,Y_2,\dots$, $X$ and $Y$ be defined on the same probability space and assume that $X_n\to X$ and $Y_n\to Y$ in probability. Show that
\begin{enumerate}[(a)]
 \item $X_n Y_n\to XY$ in probability.
 \item If almost surely $Y_n\ne 0$ and $Y\ne 0$, then $X_n / Y_n\to X/Y$ in probability.
\end{enumerate}
%
\item \textbf{(homework)} Prove that
$$
\lim_{n\to\infty} \int_0^1 \int_0^1 \dots \int_0^1 \frac{x_1^2+x_2^2+\dots x_n^2}{x_1+x_2+\dots x_n} \diff x_1 \diff x_2 \dots \diff x_n =\frac{2}{3}.
$$
%
\item Let $f:[0;1]\to \IR$ be a continuous function. Prove that
\begin{enumerate}[(a)]
 \item 
$$
\lim_{n\to\infty} \int_0^1 \int_0^1 \dots \int_0^1 f\left(\frac{x_1+x_2+\dots x_n}{n}\right) \diff x_1 \diff x_2 \dots \diff x_n =f\left(\frac{1}{2}\right).
$$
 \item 
$$
\lim_{n\to\infty} \int_0^1 \int_0^1 \dots \int_0^1 f\left((x_1 x_2 \dots x_n)^{1/n}\right) \diff x_1 \diff x_2 \dots \diff x_n =f\left(\frac{1}{e}\right).
$$
\end{enumerate}
%
\item Let the random variables $X_1,X_2,\dots,X_n,\dots$ be defined on the same probability space and let $Y_n:=\sup_{m\ge n}|X_m|$. Prove that the following two statements are equivalent:
\begin{enumerate}[(i)]
 \item $X_n\to 0$ almost surely as $n\to\infty$.
 \item $Y_n\to 0$ in probability as $n\to\infty$.
\end{enumerate}
%
\end{enumerate}

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