{"paper":{"title":"Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Mariusz Mirek","submitted_at":"2009-07-13T22:51:31Z","abstract_excerpt":"We consider the Markov chain $\\{X_n^x\\}_{n=0}^\\infty$ on $\\R^d$ defined by the stochastic recursion $X_{n}^{x}=\\p_{\\theta_{n}}(X_{n-1}^{x})$, starting at $x\\in\\R^d$, where $\\theta_{1}, \\theta_{2},...$ are i.i.d. random variables taking their values in a metric space $(\\Theta, \\mathfrak{r}),$ and $\\p_{\\theta_{n}}:\\R^d\\mapsto\\R^d$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure $\\nu$. Under appropriate assumptions on $\\p_{\\theta_n}$, we will show that the measure $\\nu$ has a heavy tail with the exponent $\\alpha>0$ i.e. $\\nu(\\{x\\in\\R^d: |x|>t\\})\\asymp t^{-\\alpha}$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.2261","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"}