Extreme values of random or chaotic discretization steps and connected networks

Dominique Guegan () and Matthieu Garcin ()
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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 nationale des ponts et chaussées - 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

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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 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.

Keywords: invariant measure; ad hoc network; Spacings; extreme value theory; copula; discretization; dynamical systems (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (8)

Published in Applied Mathematical Sciences, 2012, 6 (119), pp.5901-5926

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:halshs-00750231

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