\documentclass[12pt]{article}
\input epsf      
\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{\A}{\mathcal{A}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\E}{\mathcal{E}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\N}{\mathcal{N}}
\newcommand{\ra}{\rightarrow}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\naturals}{\mathbb{N}}
\newcommand{\complex}{\mathbb{C}}
\newcommand{\half}{\mathbb{H}}
\newcommand{\implies}{\Rightarrow}


\begin{document}

\centerline {\bf Math 523b - Homework 6 (Last one!) } 

\bigskip

\begin{enumerate}

\item Folland problem 22, p. 225

\item Folland problem 24, p. 225

\item Folland problem 27, p. 225


\item  In a homework problem you showed that the dual of $c_0$, 
the space of sequences that converge to zero with the sup norm, 
is isometric to $l^1$. Now let $c$ denoted the space of real-valued 
sequences that converge to some finite real number. 
Given $y=\{y_n\}_{n=1}^\infty \in l^1$ we can still define a linear
functional $f_y$ by 
$$
f_y(\{x_n\}_{n=1}^\infty)=\sum_{n=1}^\infty x_n y_n
$$
and the same argument as for $c_0$ proves this is a bounded linear
functional with norm equal to the $l^1$ norm of $y$. 
So we have an isometry from $l^1$ {\bf into} $c^*$.
\begin{enumerate} 
\item The Riesz representation theorem says that the dual of $c_0$ is 
isometric with a space of measures. Explain briefly how this is 
related to the assertion that $c_0^*$ is isometric with $l^1$. 
\item Prove that the isometry above from $l^1$ into $c^*$ is not onto.
\item Prove that there is a linear function $f_\infty \in c^*$ so 
that every linear functional in $c^*$ can be written as 
$f_y + a f_\infty$ for some $y \in l^1$ and some real number $a$. 
(So the range of the isometry has co-dimension 1.)
\end{enumerate} 


\item Let $f(x)$ be a Borel measurable real-valued function on $[0,1]$ 
with \\
$\int_0^1 f^2(x) \, dx < \infty$. Let $\{U_n\}_{n=1}^\infty$
be an independent sequence of random variables, each of which is uniformly 
distributed on $[0,1]$, i.e., the distribution $P_{U_N}$ of $U_n$ is Lebesgue 
measure on $[0,1]$. Let 
\bea
S_n= {1 \over n} \sum_{i=1}^n f(U_i)
\nonumber
\eea
Prove that $S_n$ converges to the constant $\int_0^1 f(x) \, dx$
in probability.


\item 
Let $E_n$ be a sequence of events. Recall that 
\bea
\limsup E_n = \cap_{n=1}^\infty \cup_{k=n}^\infty E_k
\eea
In probability this event is usually denoted $E_n \, i.o.$
where i.o. stands for ``infinitely often.''
Let $X_n$ be an independent, identically distributed 
sequence of non-negative random variables. 
Prove that $EX_1 <\infty$ if and only 
if $P(X_n \ge n \, \, i.o.)=0$. 

\item Suppose $X_n$ are independent, identically distributed, and have 
finite second moment, i.e.,
$EX_n^2 < \infty$. We proved a weak law of large numbers 
which says that for any
$\epsilon > 0$, the probability 
$$P(|{1 \over n} \sum_{k=1}^n X_k - \mu| > \epsilon)$$
converges to zero as $n \rightarrow \infty$, but we didn't say anything 
about how fast it converges.

\no (a) Use the proof from class of the weak law for the
case of finite second moment to show it converges to zero at least 
as fast as $1/n^p$ for some power $p$. (I.e., show that the probability 
is $\le c/n^p$ for some constant $c$, which can depend on $\epsilon$.)
You should find the biggest $p$ you can. 

\no (b) Now suppose that you also know that $EX_n^4 < \infty$.
Prove that $P(|{1 \over n} \sum_{k=1}^n X_k - \mu| > \epsilon)$
converges to zero faster than your bound from (a), i.e., with a bigger $p$. 


\end{enumerate}
\end{document}
\end



