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 .
Categories: math.PR