\magnification=1200
\baselineskip=20pt
\nopagenumbers
\font\big=cmr12 scaled \magstep2

\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
\bigskip

\baselineskip=12pt
\centerline{4:15 p.m., Tuesday, April 25, 2000}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

\bigskip
\baselineskip=15pt
\centerline{\sl Iain Johnstone}
\centerline{\sl Department of Statistics}
\centerline{\sl Stanford University}
\bigskip
\centerline{\bf The largest eigenvalue of a large white Wishart matrix}
\bigskip


Let $X$ be an $n \times p$ data matrix whose rows are independent draws
from $N_p (0,\Sigma).$
In applications, it is now common for $p$ as well as $n$ to be quite large.
The eigenvalues, or
principal components, of the sample covariance matrix are then known to be
more dispersed than the population eigenvalues, particularly when $\Sigma =
\sigma^2 I.$  In this latter case, the methods of random matrix theory
quantify the effect precisely: they yield the limiting distribution of the
largest
eigenvalue as $n, p \rightarrow \infty$ with $n/p = \gamma \geq 1.$ The
approximation is informative for $\min (n,p)$ as small as 10, and may
complement graphical tools, such as the screeplot, in isolating significant
principal components.

\bye