Extreme Value and Record Statistics in Heavy-Tailed Processes With Long-Range Memory

Abstract

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 heavytailed distributions, often called fractional Lévy noise. While the size distribution of extreme events is not affected by the long-range memory in the asymptotic limit and remains a Fréchet 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.

25 Years of Self-Organized Criticality: Solar and Astrophysics

“…The best relevant data we have at hand is the solar flare statistics from the last 40 years, while geological tracers (nitrate concentrations in polar ice cores or select radionuclides) extend over millenia, but are not reliable proxy records of solar flares or SEP events (Schrijver et al 2012), because nitrate spikes in ice cores can also be caused by biomass burning plumes (Wolff et al 2012). Theoretical studies focus on extreme value and record statistics in heavy-tailed processes with long-range memory (Schumann et al 2012). The inclusion of memory and persistence is obviously very important, because the predicted number of extreme events during a clustered time interval can be much larger than predicted in a purely stochastic SOC model, such as in the original BTW model (Strugarek and Charbonneau 2014).…”

Triggering cascades and statistical properties of aftershocks

“…[31] In particular, such behavior is expected based on the Omori-Utsu law and the productivity law, which in the context of the ETAS model fully determine the average time interval between a main shock and its last aftershock: From extreme value statistics it follows that the average duration of a directly triggered aftershock sequence scales asymptotically as 10 (˛/p)m [Schumann et al, 2012]. Figure 11 shows that our method to define aftershocks allows us to roughly recover such a scaling for the ETAS model.…”

Solar cycle variation of the statistical distribution of the solar wind ε parameter and its constituent variables

“…Moloney and Davidsen [] found that the block maxima of the ε parameter follow the Fréchet distribution; however, this fit overestimated the highest values. Since the Fréchet class is the limiting distribution for a variable that has a PDF with a power law tail [ Schumann et al , ], a Fréchet fit to block maxima of the ε parameter would support Burlaga's earlier proposal of a lognormal PDF with a Pareto tail. Burlaga and Lazarus [] also found evidence for lognormal distributions of the solar wind speed, density, and temperature, and that these distributions vary with the phase of the cycle.…”