Let $X$ be a nonstationary Gaussian process, asymptotically
  centered with constant variance. We study $U_u(t)$ the number of
  up-crossings of level $u$ by the process $X$ on the interval
  $(0,t]$. Under some conditions it is shown that the point process
  $U_u(\cdot)$ converges weakly, after renormalization, to a standard
  Poisson process as $u$ tends to infinity. This implies weak
  convergence of the normalized maximum to the extreme Gumbel
  distribution.


Joint work with Jean-Marc Azai