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.
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.
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@article{CRMATH_2013__351_17-18_703_0, author = {Guivarc'h, Yves and Le Page, \'Emile}, 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}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2013.09.017/} }
TY - JOUR AU - Guivarcʼh, Yves AU - Le Page, Émile TI - Asymptotique des valeurs extrêmes pour les marches aléatoires affines JO - Comptes Rendus. Mathématique PY - 2013 SP - 703 EP - 705 VL - 351 IS - 17-18 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2013.09.017/ DO - 10.1016/j.crma.2013.09.017 LA - fr ID - CRMATH_2013__351_17-18_703_0 ER -
%0 Journal Article %A Guivarcʼh, Yves %A Le Page, Émile %T Asymptotique des valeurs extrêmes pour les marches aléatoires affines %J Comptes Rendus. Mathématique %D 2013 %P 703-705 %V 351 %N 17-18 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2013.09.017/ %R 10.1016/j.crma.2013.09.017 %G fr %F CRMATH_2013__351_17-18_703_0
Guivarcʼh, Yves; Le Page, Émile. 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. http://archive.numdam.org/articles/10.1016/j.crma.2013.09.017/
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