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

Given a sequence $(M_n,Q_n{)}_{n\ge 1}$ of i.i.d.\ random variables with generic copy $(M,Q)\in GL(d,\mathbb{R})\times {\mathbb{R}}^d$, 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 $$\underset{n\to \mathrm{\infty }}{\lim }(\text{E}\text{norm}{M_1\mathrm{.}\mathrm{.}\mathrm{.}{M}_n}^{\kappa }{)}^{\frac{1}{n}}=1.$$ We prove, under suitable assumptions, that the sequence $S_n=R_1+\mathrm{.}\mathrm{.}\mathrm{.}+R_n$, 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 .