

\subsection{\bf Congruential Random Numbers}
\subsection{\bf in Monte Carlo}

\begin{slide}
\begin{itemize}
\item If the cycle has a length $m$, then imagine our calculation of
$\pi$
\[
\pi = \int\limits^{+1}_{-1} \int\limits^{+1}_{-1} dx\, dy\, \theta\, 
\lbrack 1 - x^2 - y^2 \rbrack
\]

\item Then random numbers after the $m$'th are no longer independent
and so our use of central limit theorem breaks down.

\item Our error is $\propto \max \left\lbrack \frac{1}{\sqrt{N}} ,
\frac{1}{\sqrt{m}} \right\rbrack$ where $N= $ number of events and $m$
is cycle length.

\item The references give various formal criteria to guide the choice
of $a$, $c$, and $m$.  

\item Unfortunately, I do not see any of these references as being very
good as they do not test and compare some key choices.
\end{itemize}
\end{slide}

