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Valeurs extrêmes de mosaïques aléatoires

Abstract : A random tessellation is a partition of the Euclidean space into polytopes that are called cells. Such a structure appears in many domains such as cellular biology, telecommunications and image segmentation. Many results were established on the typical cell i.e. a cell which is ''chosen uniformly'' in the tessellation. Nevertheless, these works do not reflect the regularity of the tessellation and the pathology of several cells (e.g. elongated or big cells). In this PhD thesis, we investigate the random tessellations by a new approach which is Extreme Value Theory. In practice, we observe the random tessellation in a window and we consider a geometrical characteristic (e.g. the volume, the number of vertices or the diameter of the cells). Our problem is to investigate the behaviour of the maximum and minimum (and more generally the order statistics) of this characteristic for the cells of the window when the size of the window tends to infinity. Such an approach leads to a better description of the regularity of the tessellation. It provides also some tools to investigate the quality of a discrete approximation between a set and the cells of a random tessellation. Another potential application field is the statistics of point processes. Our results concern mainly limit theorems on the extremes and order statistics of various geometrical characteristics and random tessellations. In particular, we provide the rates of convergence with some delicate geometric estimates. We derive an upper bound of the Hausdorff distance between a set and its so-called Poisson-Voronoi approximation from the investigation of the maximum of diameters. Besides, we deal with geometrical aspects such as boundary effects and shape of the optimizing cells. Finally, in order to study the repartition of the exceedance cells (i.e. cells with a large characteristic), we are interested by the convergence of underlying point processes and by the mean size of a cluster of exceedances. Our tools come from stochastic geometry (Palm measure, Slivnyak's formula, covering probabilities) and Extreme Value Theory (dependency graphs, Chen-Stein method, extremal index).
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Submitted on : Friday, December 13, 2013 - 3:33:15 AM
Last modification on : Tuesday, October 19, 2021 - 4:13:29 PM
Long-term archiving on: : Friday, March 14, 2014 - 12:45:27 AM

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  • HAL Id : tel-00918145, version 1

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Nicolas Chenavier. Valeurs extrêmes de mosaïques aléatoires. Probabilités [math.PR]. Université de Rouen, 2013. Français. ⟨tel-00918145⟩

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