On fixed points of a generalized multidimensional affine recursion

Let G be a multiplicative subsemigroup of the general linear group Gl ${(\mathbb{R}^d)}$ 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 $$R\stackrel{\mathcal{D}}{=} \sum_{i=1}^N A_iR_i+B,$$ where N ≥ 2 is a fixed natural number, A ₁, . . . , A _N are independent copies of ${A, B \in \mathbb{R}^d}$ is a random vector with positive entries, and R ₁, . . . , R _N are independent copies of ${R \in \mathbb{R}^d}$ , which have also positive entries. Moreover, all of them are mutually independent and ${\stackrel{\mathcal{D}}{=}}$ stands for the equality in distribution. We will show with the aid of spectral theory developed by Guivarc’h and Le Page (Simplicité de spectres de Lyapounov et propriété d’isolation spectrale pour une famille d’opérateurs de transfert sur l’espace projectif. Random Walks and Geometry, Walter de Gruyter GmbH & Co. KG, Berlin, 2004; On matricial renewal theorems and tails of stationary measures for affine stochastic recursions, Preprint, 2011) and Kesten’s renewal theorem (Kesten in Ann Probab 2:355–386, 1974), that under appropriate conditions, there exists χ >  0 such that ${{\mathbb{P}(\{\langle R, u \rangle > t\})\asymp t^{-\chi}}}$ , as t → ∞, for every unit vector ${u \in \mathbb{S}^{d-1}}$ with positive entries.