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Extreme values of random or chaotic discretization steps

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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 a phone 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

  • Matthieu Garcin & Dominique Guegan, 2012. "Extreme values of random or chaotic discretization steps," Documents de travail du Centre d'Economie de la Sorbonne 12033, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
  • Handle: RePEc:mse:cesdoc:12033
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    1. Blundell, Richard & Bond, Stephen, 1998. "Initial conditions and moment restrictions in dynamic panel data models," Journal of Econometrics, Elsevier, pages 115-143.
    2. Yair Mundlak, 1961. "Empirical Production Function Free of Management Bias," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 43(1), pages 44-56.
    3. James Levinsohn & Amil Petrin, 2003. "Estimating Production Functions Using Inputs to Control for Unobservables," Review of Economic Studies, Oxford University Press, vol. 70(2), pages 317-341.
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    Cited by:

    1. Matthieu Garcin & Dominique Guegan & Bertrand Hassani, 2017. "A novel multivariate risk measure: the Kendall VaR," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01467857, HAL.
    2. repec:eee:phsmap:v:483:y:2017:i:c:p:462-479 is not listed on IDEAS
    3. 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.

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    Keywords

    Spacings; extreme values; copula; discretization; dynamical systems; invariant measure;

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