On fixed points of a generalized multidimensional affine recursion

TL;DR

This paper investigates the existence and properties of fixed points for a class of multidimensional affine recursions driven by random matrices with positive entries, using spectral theory and renewal theorems.

Contribution

It establishes conditions under which a unique fixed point exists and characterizes its tail behavior in multidimensional affine stochastic equations.

Findings

Existence of a fixed point with heavy-tailed distribution.

Asymptotic tail behavior follows a power law with exponent hi.

Utilizes spectral theory and renewal theorems for analysis.

Abstract

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 $N\ge2$ 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 and Kesten's renewal theorem, that under appropriate conditions, there exists $\chi>0$ such that $\P(\{<R, u>>t\})\asymp t^{-\chi},$ as $t\to\8$, for every unit vector $u\in\mathbb{S}^{d-1}$ with positive entries.