  We consider the extreme value statistics of correlated random variables that
arise from a Langevin equation. Recently, it was shown that the extreme values
of the Ornstein-Uhlenbeck process follow a different distribution than those
originating from its equilibrium measure, composed of independent and
identically distributed Gaussian random variables. Here, we first focus on the
discretely sampled Ornstein-Uhlenbeck process, which interpolates between these
two limits. We show that in the limit of large times, its extreme values
converge to those of the equilibrium distribution, instead of those of the
continuously sampled process. This finding folds for any positive sampling
interval, with an abrupt transition at zero. We then analyze the Langevin
equation for any force that gives rise to a stable equilibrium distribution.
For forces which asymptotically grow with the distance from the equilibrium
point, the above conclusion continues to hold, and the extreme values for large
times correspond to those of independent variables drawn from the equilibrium
distribution. However, for forces which asymptotically decay to zero with the
distance, the discretely sampled extreme value statistics at large times
approach those of the continuously sampled process.
