  Let $M_n$ be the maximum of $n$ zero-mean gaussian variables $X_1,..,X_n$
with covariance matrix of minimum eigenvalue $\lambda$ and maximum eigenvalue
$\Lambda$. Then, for $n \ge 70$, $$\Pr\{M_n \ge \lambda \left (2 \log n - 2.5 -
\log(2 \log n - 2.5) \right )^\frac{1}{2} -.68\Lambda\} \ge \frac{1}{2}.$$
Bounds are also given for tail probabilities other than $\frac{1}{2}$. Upper
bounds are given for tail probabilities of the maximum of dependent identically
distributed variables. As an application, the maximum of purely
non-deterministic stationary Gaussian processes is shown to have the same first
order asymptotic behaviour as the maximum of independent gaussian processes.
