Convergence to stable laws for multidimensional stochastic recursions: the case of regular matrices

TL;DR

This paper proves that sums of solutions to a multidimensional stochastic recursion with regular matrices converge to a stable law, and characterizes the tail behavior of the stationary solution.

Contribution

It extends previous results by establishing convergence to stable laws for multidimensional recursions with regular matrices and describes the tail measure of the stationary solution.

Findings

Sum of recursive solutions converges to a stable distribution.

Stationary solution exhibits regular variation with index κ.

Provides a detailed description of the tail measure.

Abstract

Given a sequence $(M_{n},Q_{n})_{n\ge 1}$ of i.i.d.\ random variables with generic copy $(M,Q) \in GL(d, \R) \times \R^d$, we consider the random difference equation (RDE) $$ R_{n}=M_{n}R_{n-1}+Q_{n}, $$ $n\ge 1$, and assume the existence of $\kappa >0$ such that $$ \lim_{n \to \infty}(\E{\norm{M_1 ... M_n}^\kappa})^{\frac{1}{n}} = 1 .$$ We prove, under suitable assumptions, that the sequence $S_n = R_1 + ... + R_n$, appropriately normalized, converges in law to a multidimensional stable distribution with index $\kappa$. As a by-product, we show that the unique stationary solution $R$ of the RDE is regularly varying with index $\kappa$, and give a precise description of its tail measure. This extends the prior work http://arxiv.org/abs/1009.1728v3 .