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\begin{document}
\begin{center}
{\bf Probability 1\\
CEU Budapest, fall semester 2013}\\
Imre Péter Tóth\\
{\bf Homework sheet 3 -- due on 07.10.2013 -- and exercises for practice}\\
\end{center}


\begin{enumerate}[3.1]
%
\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.
%
\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 \emph{Exchangeability of integral and limit.} Consider the sequences of functions $f_n:[0,1]\to \IR$ and $g_n:[0,1]\to \IR$ concerning their pointwise limits and the limits of their integrals. Do there exist integrable functions $f:[0,1]\to \IR$ and $g:[0,1]\to \IR$, such that $f_n(x)\to f(x)$ and $g_n(x)\to g(x)$ for Lebesgue almost every $x\in[0,1]$? What is $\li \left( \int\limits_0^1 f_n(x) dx \right)$ and $\li \left( \int\limits_0^1 g_n(x) dx \right)$? Are the conditions of the dominated and monotone convergence theorems and the Fatou lemma satisfied? If yes, what do these theorems ensure about these specific examples?
\begin{enumerate}
\item
\[ f_n(x)=
    \begin{cases}
    n^2x  & \text{ if } 0\le x<1/n, \\
    2n-n^2 x  & \text{ if } 1/n\le x \le 2/n, \\
    0& \text{ otherwise}. \\
    \end{cases}\]
\item Write $n$ as $n=2^k+l$, where $k=0,1,2\dots$ and $l=0,1,\dots,2^k-1$ (this can be done in a unique way for every  $n$). Now let
\[ g_n(x)=
    \begin{cases}
    1  & \text{ if } \frac{l}{2^k} \le x <\frac{l+1}{2^k}, \\
    0& \text{ otherwise}. \\
    \end{cases}\]
\end{enumerate}
%
\item \textbf{(homework)} \emph{Exchangeability of integrals.} Consider the following function $f:\IR^2\to\IR$:
\[ f(x)=
    \begin{cases}
    1  & \text{ if } \quad 0< x,\, 0<y \text{ and } 0\le x-y\le 1,  \\
    -1  & \text{ if } \quad 0< x,\, 0<y \text{ and } 0< y-x\le 1, \\
    0& \text{ otherwise}. \\
    \end{cases}\]
Calculate $\int\limits_{-\infty}^{+\infty} \left(\int\limits_{-\infty}^{+\infty} f(x,y)dx \right) dy$ and $\int\limits_{-\infty}^{+\infty} \left(\int\limits_{-\infty}^{+\infty} f(x,y)dy \right) dx$. What's the situation with the Fubini theorem?
%
\item \emph{Weak convergence and densities.}
\begin{enumerate}
 \item  \textbf{(homework)} Prove the following
 \begin{theorem}
  Let $\mu_1,\mu_2,\dots$ and $\mu$ be a sequence of probability distributions on $\IR$ which are absolutely continouos w.r.t. Lebesgue measure. Denote their densities by $f_1,f_2,\dots$ and $f$, respectively. Suppose that $f_n(x)\stackrel{n\to\infty}{\longrightarrow} f(x)$ for every $x\in\IR$. Then $\mu_n\Rightarrow \mu$ (weakly).
 \end{theorem}
 (Hint: denote the cumulative distribution functions by $F_1,F_2,\dots$ and $F$, respectively. Use the Fatou lemma to show that $F(x)\le\liminf_{n\to\infty}F_n(x)$. For the other direction, consider $G(x):=1-F(x)$.
 \item Show examples of the following facts:
  \begin{enumerate}
   \item It can happen that the $f_n$ converge pointwise to some $f$, but the sequence $\mu_n$ is not weakly convergent, because $f$ is not a density.
   \item It can happen that the $\mu_n$ are absolutely continuous, $\mu_n\Rightarrow \mu$, but $\mu$ is not absolutely continuous.
   \item It can happen that the $\mu_n$ and also $\mu$ are absolutely continuous, $\mu_n\Rightarrow \mu$, but $f_n(x)$ does not converge to $f(x)$ for any $x$.
  \end{enumerate}
\end{enumerate}
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\end{enumerate}

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