The paper considers a Markov chain \(\{X_n\}_{n = 0}^\infty\) on \(\mathbb R^d\) defined by the stochastic recursion \(X_n= M_nX_{n-1}+ Q_n\), where \((Q_n,M_n)\) are i.i.d. random variables taking values in the affine group \(\mathbb R^d\times \text{GL}(\mathbb R^d)\). For the main part of the paper \({M_n}\) is assumed to belong to the similarity group of \(\mathbb R^d\). The authors investigate the weak convergence of the properly centered and normalized partial sums \(S_n= \sum_{k=1}^n {X_k}\) to a normal law or to an infinitely divisible law, which is stable in a natural sense. A local limit theorem for the sums \(S_n\) is proved under an aperiodicity hypothesis.