\documentclass[12pt]{article}
\usepackage{graphicx}% Include figure files
\usepackage{amssymb}
\usepackage{amsfonts}
%\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}

\newcommand{\ba}[1]{\begin{array}{#1}}
\newcommand{\ea}{\end{array}}
\newcommand{\arr}[2]{\ba{#1} #2 \ea}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray*}}
\newcommand{\eea}{\end{eqnarray*}}
\newcommand {\no}  {\noindent}
\newcommand{\borel}{\mathcal{B}}
\newcommand{\f}{\mathcal{F}}
\newcommand{\e}{\mathcal{E}}
\newcommand{\ra}{\rightarrow}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\naturals}{\mathbb{N}}

\begin{document}

\centerline {\bf Math 563 - Fall '14 - Homework 6} 

\smallskip
  
{\bf Do 6 of the 7 problems} 

\smallskip

\no 1. The Cauchy distribution has the density function
$$
f(x) =\frac{1}{\pi (1+x^2)}
$$
I.e., the distribution is Lebesgue measure on the real line times
this function.

\no (a) Compute the characteristic function of a Cauchy random variable. 
Hint: contour integration.

\no (b) Show that if $X_n$ is an i.i.d. sequence with the Cauchy distribution,
then for all $n$, $\frac{1}{n}\sum_{k=1}^n X_k$ has the Cauchy distribution. 

\medskip

\no 2. Suppose that $X_n$ converges in distribution to $X$ and $Y_n$ 
converges in distribution to $Y$. Suppose further that for each $n$,
$X_n$ and $Y_n$ are independent, and that $X$ and $Y$ are independent. 
Assume that $X_n, Y_n, X, Y$ are defined on the same
probability space.  Prove that $X_n + Y_n$ converges in distribution to 
$X+Y$. 


\medskip

\no 3. (from Durrett) Let $X_n$ be an iid sequence such that for all $n$, 
$X_n \ge 0$, $E X_n=1$, $var(X_n)=\sigma^2$ where $\sigma \in (0,\infty)$. 
Let $S_n=\sum_{k=1}^n X_k$. Prove that 
$\sqrt{S_n}-\sqrt{n}$ converges in distribution to a normal distribution and 
find the mean and variance of the limiting normal distribution.

\medskip

\no 4. Let $E_n$ be independent events such that
$\sum_n P(E_n) = \infty$ and $P(E_n) \ra 0$. Prove there are  
increasing sequences $c_n$ and $d_n$, both of which 
go to $\infty$ such that
$$
\frac{\sum_{k=1}^n 1_{E_k} - c_n}{d_n} \Rightarrow Z
$$
where $Z$ is a random variable with the standard normal distribution.

\medskip

\no 5. Let $X$ be a real valued random variable. Let $\phi(t)$ be 
its characteristic function. 

\smallskip

\no (a) Show that if there is an $a \ne 0$ such 
that $\phi(2 \pi a) = 1$, then $aX$ takes values in the integers a.s.

\no (b) Show that if there is an $a \ne 0$ such 
that $|\phi(2 \pi a)| = 1$, then there is a real constant $b$ 
such that $aX+b$ takes values in the integers a.s.

\no (c) Show that if there is an interval $(a,b)$ such that 
$|\phi(t)|=1$ for $t \in (a,b)$, then $X$ is a constant a.s.  


\medskip

\no 6. Let $X_n$ be an i.i.d. sequence with $E X_n=\mu$ and 
$Var(X_n)=\sigma^2 < \infty$. 
The sample mean is defined by 
$$
\overline{X}_n = \frac{1}{n} \sum_{k=1}^n X_k
$$
By the strong law of large numbers it converges a.s. to $\mu$. 
The sample variance is defined by 
$$
V_n = \frac{1}{n-1} \sum_{k=1}^n (X_k - \overline{X}_n)^2
$$

\no (a) Prove that $V_n$ converges a.s. to $\sigma^2$. 
Hint: the variance is defined as $E[(X-\mu)^2]$. It is also equal to 
$E[X^2] - \mu^2$. The sample variance is almost equal to 
$$
\frac{1}{n} \sum_{k=1}^n X_k^2 - \overline{X}_n^2
$$

\no (b) (loosely based on exercise 4.5, ``self-normlized sums'', in Durrett)
Prove that 
\bea
\frac{\sum_{k=1}^n X_k - n \mu}{\sqrt{n V_n}}
\eea
converges in distribution to the standard normal distribution.

\medskip

\no 7. The gamma distribution is a two parameter family of densities.
The parameters $\lambda$ and $w$ are both positive.  
The density is 
$$
f(x)=\frac{\lambda^w}{\Gamma(w)} x^{w-1} \exp(-\lambda x) 
$$ 
for $x \ge 0$. The density is $0$ for $x<0$.
The gamma function is defined by 
$$
\Gamma(w) = \int_0^\infty \, x^{w-1} \, e^{-x} \, dx 
$$
Integration by parts shows $\Gamma(w+1)=w \Gamma(w)$. If $w$ is 
an integer then $\Gamma(w)=(w-1)!$. A little calculus shows the mean 
of the gamma distribution is $w/\lambda$ and the variance is $w/\lambda^2$. 
For positive integers $n$ let $X_n$ be a random variable with a 
gamma distribution with $w=n$ and $\lambda=1$. Prove that
$(X_n - n)/\sqrt{n}$ converges in distribution to a standard normal. 

\end{document}
\end


