Discrete sampling of correlated random variables modifies the long-time behavior of their extreme value statistics

TL;DR

This paper investigates how discrete sampling affects the long-time extreme value statistics of correlated Langevin processes, revealing a transition between equilibrium and continuous sampling behaviors depending on the force and sampling interval.

Contribution

It demonstrates that discretely sampled correlated variables exhibit a transition in extreme value behavior, bridging the gap between independent and correlated cases, with implications for stochastic process analysis.

Findings

Large-time extremes of discretely sampled processes match equilibrium distribution for growing forces.

For forces decaying to zero, discrete sampling results align with continuous sampling extremes.

An abrupt transition occurs at zero sampling interval, affecting the extreme value distribution.

Abstract

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.