\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{epsfig}

\def\Q{{\mathbb Q}}        % rationals
\def\Z{{\mathbb Z}}        % integers
\def\R{{\mathbb R}}        % reals
\def\Rn{{\R^{n}}}          % product of n copies of reals

\def\X{{\Omega}}            % State space
\def\P{{\mathbb P}}        % probability
\def\E{{\mathbb E}}        % expectation
\def\1{{\mathbf 1}}        % indicator
\def\var{{\mathop{\mathbf Var}}}    % variance
\def\ep{\epsilon}

\def\eqd{\stackrel{\scriptscriptstyle d}{=}}
\def\deq{{:=}}
\def\borel{{\cal B}}    % borel sigmal field
\def\G{{\cal G}}        % some sigma field
\def\F{{\cal F}}        % some sigma field
\def\H{{\cal H}}        % some sigma field
\def\Pm{{\mathrm{P}}}
\def\Qm{{\mathrm{Q}}}
\def\L{{\mathbf L}}     % L, as in L^2
\def\coupletime{{\tau_{couple}}}
\def\mtv{{\overline{d}}}
\def\one{{\mathbf{1}}}

\def\ascv{\stackrel{\scriptscriptstyle a.s.}{\longrightarrow}}     % almost sure convergnece
\def\pcv{\stackrel{\scriptscriptstyle \P}{\longrightarrow}}        % convergnece in P
\def\ltcv{\stackrel{\scriptscriptstyle\L^2}{\longrightarrow}}      % L2 convergnece
\def\lpcv{\stackrel{\scriptscriptstyle\L^p}{\longrightarrow}}      % Lp convergnece

\def\ci{\perp\!\!\!\perp}  % conditional independence

% the header/lecture structure...borrowed from alistair sinclair's cs270 format
\newcommand{\lecture}[4]{
   \pagestyle{myheadings}
   \thispagestyle{plain}
   \newpage
   \setcounter{lecnum}{#1}
   \setcounter{page}{1}
   \noindent
   \begin{center}

   \framebox{
     \vbox{\vspace{2mm}
       \hbox to 5.90in { \hfill {\LARGE Lecture #1 : #2} \hfill }
       \vspace{8mm}
       \hbox to 5.90in {
                         \hfill Lecturer: Yuval Peres \,
Scribe: {\it #3} {\footnotesize{\tt <#4>}\hfill }
                       }
      \vspace{2mm}}
   }
   \end{center}
   \markboth{Lecture #1: #2}{Lecture #1: #2}
   \vspace*{4mm}
}
       %\hbox to 5.90in { \hfill {\it Scribe: #3} }

% number everything using the lecture number counter
% so there is never any question as to whether we
% are referring to an example, theorem, exercise, etc.
\newcounter{lecnum}
\renewcommand{\thepage}{\thelecnum-\arabic{page}}
\renewcommand{\thesection}{\thelecnum.\arabic{section}}
\renewcommand{\theequation}{\thelecnum.\arabic{equation}}
\renewcommand{\thefigure}{\thelecnum.\arabic{figure}}
\renewcommand{\thetable}{\thelecnum.\arabic{table}}

\newtheorem{theorem}{Theorem}[lecnum]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
%\newenvironment{proof}{{\bf Proof:}}{\hfill\rule{2mm}{2mm}}
\newenvironment{proofsketch}{{\bf Proof Sketch:}}{\hfill\rule{2mm}{2mm}}
\newcommand{\fig}[3]{
      \vspace{#2}
      \begin{center}
      Figure \thelecnum.#1:~#3
      \end{center}
}
\newtheorem{exercise}[theorem]{Exercise}

%some things we'll never understand....alas
\setlength{\oddsidemargin}{0.25 in}
\setlength{\evensidemargin}{-0.25 in}
\setlength{\topmargin}{-0.6 in}
\setlength{\textwidth}{6.0 in}
\setlength{\textheight}{8.5 in}
\setlength{\headsep}{0.75 in}
\setlength{\parindent}{0 in}
\setlength{\parskip}{0.1 in}
\advance\topmargin by 0.5in