{"paper":{"title":"Random iteration with place dependent probabilities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.PR","authors_text":"M. \\'Sl\\k{e}czka, R. Kapica","submitted_at":"2011-07-04T19:24:03Z","abstract_excerpt":"Markov chains arising from random iteration of functions $S_{\\theta}:X\\to X$, $\\theta \\in \\Theta$, where $X$ is a Polish space and $\\Theta$ is arbitrary set of indices are considerd. At $x\\in X$, $\\theta$ is sampled from distribution $\\theta_x$ on $\\Theta$ and $\\theta_x$ are different for different $x$. Exponential convergence to a unique invariant measure is proved. This result is applied to case of random affine transformations on ${\\mathbb R}^d$ giving existence of exponentially attractive perpetuities with place dependent probabilities."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.0707","kind":"arxiv","version":3},"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"}