2011/11/07 by Mariusz Mirek, Mirek, Mariusz · 1 citation
Mathematics · #60J35 #60J80 #60K05 #Advanced Algebra and Geometry #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Random Matrices and Applications #math.PR #msc:60J35 #msc:60J80 #msc:60K05
paper · pdf · doi:10.48550/arxiv.1111.1756
30 pages, no figures
arxiv created 2011/11/07 · openalex publication_date 2011/11/07 · arxiv updated 2011/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a multiplicative subsemigroup of the general linear group \Gl(ℝd) which consists of matrices with positive entries such that every column and every row contains a strictly positive element. Given a G--valued random matrix A, we consider the following generalized multidimensional affine equation R\stackrelD=∑i=1N AiRi+B, where N≥2 is a fixed natural number, A1,...,AN are independent copies of A, B∈ℝd is a random vector with positive entries, and R1,...,RN are independent copies of R∈ℝd, which have also positive entries. Moreover, all of them are mutually independent and \stackrelD= stands for the equality in distribution. We will show with the aid of spectral theory developed by Guivarc'h and Le Page and Kesten's renewal theorem, that under appropriate conditions, there exists χ>0 such that ¶(\<R, u>>t\)\asymp t-χ, as t→\8, for every unit vector u∈\mathbbSd-1 with positive entries.