{"paper":{"title":"Persistence of one-dimensional AR(1)-sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"G\\\"unter Hinrichs, Martin Kolb, Vitali Wachtel","submitted_at":"2018-01-13T20:41:24Z","abstract_excerpt":"For a class of one-dimensional autoregressive processes $(X_n)$ we consider the tail behaviour of the stopping time $T_0=\\min \\lbrace n\\geq 1: X_n\\leq 0 \\rbrace$. We discuss existing general analytical approaches to this and related problems and propose a new one, which is based on a renewal-type decomposition for the moment generating function of $T_0$ and on the analytical Fredholm alternative. Using this method, we show that $\\mathbb{P}_x(T_0=n)\\sim V(x)R_0^n$ for some $0<R_0<1$ and a positive $R^{-1}_0$-harmonic function $V$. Further we prove that our conditions on the tail behaviour of th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.04485","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"}