Nous considérons lʼespace Euclidien et une marche aléatoire affine sur , gouvernée par une probabilité λ portée par le groupe affine . Nous supposons que le sous-groupe de H engendré par le support de λ est « grand » et que la convolution par λ sur admet une unique probabilité stationnaire η dont le support est non borné. Nous montrons la convergence en loi de certains processus ponctuels associés aux valeurs extrêmes de . Les paramètres des lois limites sʼexpriment à lʼaide dʼune mesure homogène Λ sur , qui décrit lʼallure à lʼinfini de η et qui dépend essentiellement de la projection de λ sur le groupe linéaire de . En particulier, les valeurs extrêmes normalisées de suivent une loi de Fréchet, qui dépend simplement de Λ.
We consider the Euclidean space and an affine random walk on , governed by a probability λ supported on the affine group . We assume that the subgroup of H generated by the support of λ is “large” and that convolution by λ on has a unique stationary probability η such that its support is unbounded. We show the convergence in law of certain point processes associated with the extreme values of . The parameters of the limit laws are expressed in terms of a homogeneous measure Λ on , which describes the shape at infinity of η, and which depends essentially on the projection of λ on the linear group of . In particular, the normalized extreme values of follow a Fréchet law depending on Λ in a simple way.
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@article{CRMATH_2013__351_17-18_703_0,
author = {Yves Guivarc'h and \'Emile Le Page},
title = {Asymptotique des valeurs extr\^emes pour les marches al\'eatoires affines},
journal = {Comptes Rendus. Math\'ematique},
pages = {703--705},
publisher = {Elsevier},
volume = {351},
number = {17-18},
year = {2013},
doi = {10.1016/j.crma.2013.09.017},
language = {fr},
}
Yves Guivarcʼh; Émile Le Page. Asymptotique des valeurs extrêmes pour les marches aléatoires affines. Comptes Rendus. Mathématique, Volume 351 (2013) no. 17-18, pp. 703-705. doi : 10.1016/j.crma.2013.09.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.09.017/
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