2021/08/15 by Lior Zarfaty, Zarfaty, Lior, Eli Barkai +3
Economics, Econometrics and Finance · Mathematics · #Data Analysis #FOS: Physical sciences #Financial Risk and Volatility Modeling #Mathematical Physics (math-ph) #Statistical Mechanics (cond-mat.stat-mech) #Statistical Methods and Inference #Statistics and Probability (physics.data-an)
paper · pdf · doi:10.48550/arxiv.2108.06778
openalex publication_date 2021/08/15 · openalex created_date 2021/08/30 · openalex updated_date 2026/07/28
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.