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Return interval distribution of extreme events and long term memory

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  • M. S. Santhanam
  • Holger Kantz

Abstract

The distribution of recurrence times or return intervals between extreme events is important to characterize and understand the behavior of physical systems and phenomena in many disciplines. It is well known that many physical processes in nature and society display long range correlations. Hence, in the last few years, considerable research effort has been directed towards studying the distribution of return intervals for long range correlated time series. Based on numerical simulations, it was shown that the return interval distributions are of stretched exponential type. In this paper, we obtain an analytical expression for the distribution of return intervals in long range correlated time series which holds good when the average return intervals are large. We show that the distribution is actually a product of power law and a stretched exponential form. We also discuss the regimes of validity and perform detailed studies on how the return interval distribution depends on the threshold used to define extreme events.

Suggested Citation

  • M. S. Santhanam & Holger Kantz, 2008. "Return interval distribution of extreme events and long term memory," Papers 0803.1706, arXiv.org.
    • Handle: RePEc:arx:papers:0803.1706
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    Cited by:

    1. Pushpa Dissanayake & Teresa Flock & Johanna Meier & Philipp Sibbertsen, 2021. "Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights," Mathematics, MDPI, vol. 9(21), pages 1-33, November.
    2. Luciano Telesca & Zbigniew Czechowski, 2024. "Information–Theoretic Analysis of Visibility Graph Properties of Extremes in Time Series Generated by a Nonlinear Langevin Equation," Mathematics, MDPI, vol. 12(20), pages 1-15, October.
    3. Batac, Rene & Longjas, Anthony & Monterola, Christopher, 2012. "Statistical distributions of avalanche size and waiting times in an inter-sandpile cascade model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(3), pages 616-624.
    4. Xie, Wen-Jie & Jiang, Zhi-Qiang & Zhou, Wei-Xing, 2014. "Extreme value statistics and recurrence intervals of NYMEX energy futures volatility," Economic Modelling, Elsevier, vol. 36(C), pages 8-17.
    5. Li, Wei-Zhen & Zhai, Jin-Rui & Jiang, Zhi-Qiang & Wang, Gang-Jin & Zhou, Wei-Xing, 2022. "Predicting tail events in a RIA-EVT-Copula framework," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 600(C).
    6. Chicheportiche, Rémy & Chakraborti, Anirban, 2017. "A model-free characterization of recurrences in stationary time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 474(C), pages 312-318.
    7. Ribeiro, H.V. & Mendes, R.S. & Lenzi, E.K. & Belancon, M.P. & Malacarne, L.C., 2011. "On the dynamics of bubbles in boiling water," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 178-183.
    8. Zhi-Qiang Jiang & Gang-Jin Wang & Askery Canabarro & Boris Podobnik & Chi Xie & H. Eugene Stanley & Wei-Xing Zhou, 2018. "Short term prediction of extreme returns based on the recurrence interval analysis," Quantitative Finance, Taylor & Francis Journals, vol. 18(3), pages 353-370, March.
    9. Karain, Wael I., 2019. "Investigating large-amplitude protein loop motions as extreme events using recurrence interval analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 520(C), pages 1-10.
    10. A. Barbier-Chebbah & O. Bénichou & R. Voituriez & T. Guérin, 2024. "Long-term memory induced correction to Arrhenius law," Nature Communications, Nature, vol. 15(1), pages 1-7, December.
    11. Xing, Yani & Wang, Jun, 2019. "Statistical volatility duration and complexity of financial dynamics on Sierpinski gasket lattice percolation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 234-247.
    12. Czechowski, Zbigniew & Telesca, Luciano, 2024. "Effect of nonlinearity of discrete Langevin model on behavior of extremes in generated time series," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).

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