Bounding the Maximum of Dependent Random Variables

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.