  We consider a Markov chain $\{X_n\}_{n=0}^\8$ on $\R^d$ defined by the
stochastic recursion $X_{n}=M_n X_{n-1}+Q_n$, where $(Q_n,M_n)$ are i.i.d.
random variables taking values in the affine group $H=\R^d\rtimes {\rm
GL}(\R^d)$. Assume that $M_n$ takes values in the similarity group of $\R^d$,
and the Markov chain has a unique stationary measure $\nu$, which has unbounded
support. We denote by $|M_n|$ the expansion coefficient of $M_n$ and we assume
$\E |M|^\a=1$ for some positive $\a$. We show that the partial sums
$S_n=\sum_{k=0}^n X_k$, properly normalized, converge to a normal law ($\a\ge
2$) or to an infinitely divisible law, which is stable in a natural sense
($\a<2$). These laws are fully nondegenerate, if $\nu$ is not supported on an
affine hyperplane. Under a natural hypothesis, we prove also a local limit
theorem for the sums $S_n$. If $\a\le 2$, proofs are based on the homogeneity
at infinity of $\nu$ and on a detailed spectral analysis of a family of Fourier
operators $P_v$ considered as perturbations of the transition operator $P$ of
the chain $\{X_n \}$. The characteristic function of the limit law has a simple
expression in terms of moments of $\nu$ ($\a > 2$) or of the tails of $\nu$ and
of stationary measure for an associated Markov operator ($\a\le 2$). We extend
the results to the situation where $M_n$ is a random generalized similarity.
