Extreme value and record statistics in heavy-tailed processes with long-range memory

TL;DR

This paper investigates how long-range memory and heavy-tailed distributions influence extreme event statistics and record-breaking phenomena in stochastic processes, with applications to space weather and hazard assessment.

Contribution

It reveals that while the asymptotic size distribution remains Fréchet, finite-size effects and persistence significantly impact extreme event dynamics and record statistics in long-range dependent processes.

Findings

Size distribution of extremes remains Fréchet asymptotically

Finite-size effects induce strong persistence in extreme events

Long-range memory affects record-breaking statistics

Abstract

Extreme events are an important theme in various areas of science because of their typically devastating effects on society and their scientific complexities. The latter is particularly true if the underlying dynamics does not lead to independent extreme events as often observed in natural systems. Here, we focus on this case and consider stationary stochastic processes that are characterized by long-range memory and heavy-tailed distributions, often called fractional L\'evy noise. While the size distribution of extreme events is not affected by the long-range memory in the asymptotic limit and remains a Fr\'echet distribution, there are strong finite-size effects if the memory leads to persistence in the underlying dynamics. Moreover, we show that this persistence is also present in the extreme events, which allows one to make a time-dependent hazard assessment of future extreme events based on events observed in the past. This has direct applications in the field of space weather as we discuss specifically for the case of the solar power influx into the magnetosphere. Finally, we show how the statistics of records, or record-breaking extreme events, is affected by the presence of long-range memory.