{"paper":{"title":"Stability of perpetuities in Markovian environment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Fabian Buckmann, Gerold Alsmeyer","submitted_at":"2016-10-31T15:23:58Z","abstract_excerpt":"The stability of iterations of affine linear maps $\\Psi_{n}(x)=A_{n}x+B_{n}$, $n=1,2,\\ldots$, is studied in the presence of a Markovian environment, more precisely, for the situation when $(A_{n},B_{n})_{n\\ge 1}$ is modulated by an ergodic Markov chain $(M_{n})_{n\\ge 0}$ with countable state space $\\mathcal{S}$ and stationary distribution $\\pi$. We provide necessary and sufficient conditions for the a.s. and the distributional convergence of the backward iterations $\\Psi_{1}\\circ\\ldots\\circ\\Psi_{n}(Z_{0})$ and also describe all possible limit laws as solutions to a certain Markovian stochastic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.09965","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}