Asymptotique des valeurs extrêmes pour les marches aléatoires affines

Abstract

We consider the Euclidean space Rd and an affine random walk Xn on Rd, governed by a probability λ supported on the affine group H=Aff(Rd). We assume that the subgroup of H generated by the support of λ is “large” and that convolution by λ on Rd 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 Xn. The parameters of the limit laws are expressed in terms of a homogeneous measure Λ on Rd∖{0}, which describes the shape at infinity of η, and which depends essentially on the projection of λ on the linear group of Rd. In particular, the normalized extreme values of |Xn| follow a Fréchet law depending on Λ in a simple way.