A look at perpetuities via asymptotically homogeneous in space Markov chains

It is shown how a natural representation of perpetuities as asymptotically homogeneous in space Markov chains allows to prove various asymptotic tail results for stable perpetuities and limit theorems for unstable ones. Some of these results are new while others essentially improve moment conditions known in the literature. Both subexponential and Cram\'er's cases are considered.