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\begin{center}
  \textbf{\textsc{STANFORD UNIVERSITY}}\\[5pt]
  \textbf{\textsc{DEPARTMENT OF STATISTICS}}\\[5pt]
  \Large{\textbf\textsc{{DEPARTMENTAL SEMINAR}}}
\end{center}

\begin{center}
  4:15 p.m., Tuesday, June 27, 2006\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
\end{center}

\begin{center}
  \textsl{ Anthony D'Aristotile }\\
State University of New York at Plattsburgh\\
\end{center}

\begin{center}
  \textbf{ Brownian Motion and the Unitary Group    } 
\end{center}

\noindent
A theorem of Borel (1906) implies that if $\Gamma$ is chosen from
the orthogonal group $O_n$ distributed according to Haar measure and
$\gamma_n$ is any of its entries, then $\sqrt{n} \gamma_n$ converges
in distribution to a standard normal random variable as $n
\rightarrow \infty$. In joint work with Diaconis and Newman, this
result was greatly extended by showing that if $A$ is an $n \times
n$ real non-random matrix satisfying $\mbox{Tr}(AA^t) =n$, then the
trace of the matrix $A \Gamma$converges in distribution to a
standard normal random variable as   $n \rightarrow \infty$. (The
matrices $\Gamma$ and $A$ depend on n, but we have suppressed this
dependence in the notation.) This theorem is then applied to show
that appropriately normalized entries from $\Gamma$ can replace
i.i.d. standard normal random variables in the usual construction of
Brownian motion. There were also attempts to provide corresponding
theorems for the unitary group and complex Brownian motion but here
the authors were only partially successful. In this talk, we will
provide completely parallel results for the unitary group. Along the
way, we will give sufficient conditions for convergence in
distribution on the space ($D \times D$, $ \mathcal {E}) $ where $D$
is the Skorohod space of right-continuous functions on [0,1] with
left limits and $ \mathcal {E} $ is the $\sigma$-algebra of Borel
sets of $D \times D$. This extends results of Billingsley from his
book \emph{Convergence of Probability Measures}. We will also
provide the analogue for the unitary group of a result of Charles
Stein concerning the expectation of certain products of elements
from a random orthogonal matrix.

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