Discrete Sampling of Extreme Events Modifies Their Statistics

TL;DR

This paper investigates how discrete sampling affects the extreme value statistics of correlated stochastic processes modeled by Langevin dynamics, revealing that sampling rate can significantly alter EV distributions.

Contribution

It demonstrates that discrete sampling can change EV statistics from correlated to independent-like distributions depending on the potential's growth rate.

Findings

For superlinear potentials, EV distribution converges to that of i.i.d. variables.

For sublinear potentials, EV statistics match continuous sampling results.

Sampling rate critically influences the observed EV distribution.

Abstract

Extreme value (EV) statistics of correlated systems are widely investigated in many fields, spanning the spectrum from weather forecasting to earthquake prediction. Does the unavoidable discrete sampling of a continuous correlated stochastic process change its EV distribution? We explore this question for correlated random variables modeled via Langevin dynamics for a particle in a potential field. For potentials growing at infinity faster than linearly and for long measurement times, we find that the EV distribution of the discretely sampled process diverges from that of the full continuous dataset and converges to that of independent and identically distributed random variables drawn from the process's equilibrium measure. However, for processes with sublinear potentials, the long-time limit is the EV statistics of the continuously sampled data. We treat processes whose equilibrium measures belong to the three EV attractors: Gumbel, Fr\'echet, and Weibull. Our work shows that the EV statistics can be extremely sensitive to the sampling rate of the data.