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\newcommand{\indep}{\perp \!\!\! \perp} % independence
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\begin{document}
\lecture{Lecture Topic Goes Here} % Replace by the topic of the class you're taking notes for!
{\href{https://pi.math.cornell.edu/~levine/}{Lionel Levine}} % Instructor's name
{September 9, 2024} % Date of the class you're taking notes for
{Your Name Here} % Notetaker's name


\section{Brownian motion appears everywhere}

Brownian motion appears everywhere, even in this template:

\begin{theorem}
Let $B$ be a standard Brownian motion, and let $a >0$. Then
	\[ (B_t)_{t \geq 0} \stackrel{d}{=} (a^{-1/2} B_{at})_{t \geq 0} \]
\end{theorem}

\begin{proof}
They're both mean zero Gaussian processes with continuous sample paths and covariance $E[B_s B_t] = \min(s,t)$.
\end{proof}

\red{Wait, why does that prove it?}

\begin{lemma}
The law of a Gaussian process with continuous sample paths is determined by its means and covariances.
\end{lemma}

\begin{example}
If $X,Y$ are independent $N(0,1)$ then so are $\frac{X+Y}{\sqrt{2}}$, $\frac{X-Y}{\sqrt{2}}$.
\end{example}

\blue{Remind me what a Gaussian process is?}

\begin{definition}
Let $I$ be an arbitrary index set. A stochastic process $(X_i)_{i \in I}$ is a \define{Gaussian process} if every finite linear combination  
	\[ a_1 X_{i_1} + \ldots + a_n X_{i_n} \] 
has a normal distribution for all $a_1, \ldots, a_n \in \R$ and all $i_1, \ldots, i_n \in I$.
\end{definition}

\section{Insert your notes here!}

\ignore{This sentence won't appear in the pdf output.} % neither will anything preceded by a percent sign

% Now remove the stuff about Brownian motion, and replace it with your notes.  Go to it, notetaker!

\end{document}
