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\begin{document}
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\large



\title{Math 533, Spring 2020, HW10} 
\author{Due in class, April 27}
\maketitle


\begin{problem}
 If $0 < p < 1$, let $\beta_p = p \delta_1 + (1-p)\delta_0$, and let $\beta_p^{(n)}$ be the $n$th convolution power of $\beta_p$,
 \[
  \beta_p^{(n)} = \sum_{k=0}^n \frac{n!}{k!(n-k)!} p^k (1-p)^{n-k} \delta_k.
 \]
 If $a>0$, let
 \[
  \lambda_a = e^{-a} \sum_0^\infty \frac{a^k}{k!}\delta_k.
 \]
$\lambda_a$ is called the \emph{Poisson distribution} with parameter $a$.
Prove the following.
\begin{enumerate}
 \item The mean and variance of $\lambda_a$ are both $a$.
 \item $\lambda_a * \lambda_b = \lambda_{a+b}$.
 \item $\beta_{a/n}^{(n)}$ converges vaguely to $\lambda_a$ as $n \to \infty$.
\end{enumerate}

\end{problem}

\begin{problem}
 If $\sum_1^\infty n^{-2}\sigma_n^2 < \infty$, then $\lim n^{-2} \sum_1^n \sigma_j^2 = 0$. 
 If $\{a_n\} \subset \bC$ and $\lim a_n = a$, then $\lim n^{-1} \sum_1^n a_j=a$.
\end{problem}

\begin{problem}
 If $\{X_n\}$ is a sequence of independent random variables such that $\E(X_n) = 0$ and $\sum_1^\infty \sigma^2(X_n) < \infty$, then $\sum_1^\infty X_n$ converges almost surely.
\end{problem}

\begin{problem}
 If $\{X_n\}$ is a sequence of i.i.d.\ random variables which are not in $L^1$, then $\limsup n^{-1} |\sum_1^n X_j| = \infty$ almost surely.
\end{problem}

\begin{problem}(Shannon's Theorem)
 Let $\{X_i\}$ be a sequence of independent random variables on the sample space $\Omega$ having the common distribution $\lambda = \sum_1^r p_j \delta_j$ where $0 < p_j < 1$, $\sum_1^r p_j = 1$, and $\delta_j$ is the point mass at $j$.  Define random variables $Y_1, Y_2, ...$ on $\Omega$ by
 \[
  Y_n(\omega) = P\left(\left\{\omega': X_j(\omega') = X_j(\omega) \text{ for } 1 \leq j \leq n \right\} \right).
 \]
 Prove the following.
\begin{enumerate}
 \item $Y_n = \prod_1^n p_{X_j}$.
 \item $n^{-1} \log Y_n \to \sum_1^r p_j \log p_j$ almost surely.
\end{enumerate}

 
\end{problem}



\end{document}

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