  Numerical and analytical results are presented for the maximal relative
height distribution of stationary periodic Gaussian signals (one dimensional
interfaces) displaying a 1/f^alpha power spectrum. For 0<alpha<1 (regime of
decaying correlations), we observe that the mathematically established limiting
distribution (Fisher-Tippett-Gumbel distribution) is approached extremely
slowly as the sample size increases. The convergence is rapid for alpha>1
(regime of strong correlations) and a highly accurate picture gallery of
distribution functions can be constructed numerically. Analytical results can
be obtained in the limit alpha -> infinity and, for large alpha, by
perturbation expansion. Furthermore, using path integral techniques we derive a
trace formula for the distribution function, valid for alpha=2n even integer.
  From the latter we extract the small argument asymptote of the distribution
function whose analytic continuation to arbitrary alpha > 1 is found to be in
agreement with simulations. Comparison of the extreme and roughness statistics
of the interfaces reveals similarities in both the small and large argument
asymptotes of the distribution functions.
