Homogeneity at Infinity of Stationary Solutions of Multivariate Affine Stochastic Recursions

Abstract

We consider a d-dimensional affine stochastic recursion of general type corresponding to the relation, Under natural conditions, this recursion has a unique stationary solution R, which is unbounded. If d > 2, we sketch a proof of the fact that R belongs to the domain of attraction of a stable law which depends essentially of the linear part of the recursion. The proof is based on renewal theorems for products of random matrices, radial Fourier analysis in the vector space ℝd, and spectral gap properties for convolution operators on the corresponding projective space. We state the corresponding simpler result for d = 1. © Springer-Verlag Berlin Heidelberg 2013.