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Extreme values of random or chaotic discretization steps and connected networks

Author

Listed:
  • Dominique Guegan

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Matthieu Garcin

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

Abstract

By sorting independent random variables and considering the difference between two consecutive order statistics, we get random variables, called steps or spacings, that are neither independent nor identically distributed. We characterize the probability distribution of the maximum value of these steps, in three ways : i/with an exact formula ; ii/with a simple and finite approximation whose error tends to be controlled ; iii/with asymptotic behavior when the number of random variables drawn (and therefore the number of steps) tends towards infinity. The whole approach can be applied to chaotic dynamical systems by replacing the distribution of random variables by the invariant measure of the attractor when it is set. The interest of such results is twofold. In practice, for example in the telecommunications domain, one can find a lower bound for the number of antennas needed in an ad hoc network to cover an area. In theory, our results take place inside the extreme value theory extended to random variables that are neither independent nor identically distributed.

Suggested Citation

  • Dominique Guegan & Matthieu Garcin, 2012. "Extreme values of random or chaotic discretization steps and connected networks," PSE-Ecole d'économie de Paris (Postprint) halshs-00750231, HAL.
    • Handle: RePEc:hal:pseptp:halshs-00750231
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    Citations

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    Cited by:

    1. Matthieu Garcin & Dominique Guegan & Bertrand Hassani, 2018. "A novel multivariate risk measure: the Kendall VaR," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01467857, HAL.
    2. Matthieu Garcin & Dominique Guegan, 2013. "Probability density of the wavelet coefficients of a noisy chaos," Post-Print hal-00800997, HAL.
    3. Matthieu Garcin & Dominique Guegan & Bertrand Hassani, 2018. "A novel multivariate risk measure: the Kendall VaR," Post-Print halshs-01467857, HAL.
    4. Matthieu Garcin & Dominique Guegan, 2013. "Probability density of the wavelet coefficients of a noisy chaos," Documents de travail du Centre d'Economie de la Sorbonne 13015, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    5. Matthieu Garcin & Maxime L. D. Nicolas, 2024. "Nonparametric estimator of the tail dependence coefficient: balancing bias and variance," Statistical Papers, Springer, vol. 65(8), pages 4875-4913, October.
    6. Matthieu Garcin & Maxime L. D. Nicolas, 2021. "Nonparametric estimator of the tail dependence coefficient: balancing bias and variance," Papers 2111.11128, arXiv.org, revised Jul 2023.
    7. Matthieu Garcin & Dominique Guegan & Bertrand Hassani, 2017. "A novel multivariate risk measure: the Kendall VaR," Documents de travail du Centre d'Economie de la Sorbonne 17008, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    8. Matthieu Garcin & Dominique Guegan & Bertrand Hassani, 2017. "A novel multivariate risk measure: the Kendall VaR," Documents de travail du Centre d'Economie de la Sorbonne 17008r, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne, revised Apr 2018.

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