Given a sequence (M n , Q n ) n ≥ 1 of i.i.d. random variables with generic copy (M, Q) ∈ GL(d, ℝ) ×ℝ d , we consider the random difference equation (RDE) $$ R_{n}=M_{n}R_{n-1}+Q_{n}, $$ n ≥ 1, and assume the existence of κ > 0 such that $$ \lim_{n \to \infty} \left(\mathbb{E}\ensuremath{\left\| {M_1 \cdots M_n}^\kappa \right\|}\right)^{\frac{1}{n}} = 1 . $$ 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 κ. As a by-product, we show that the unique stationary solution R of the RDE is regularly varying with index κ, and give a precise description of its tail measure.