{"paper":{"title":"Null-recurrence and transience of random difference equations in the contractive case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander Iksanov, Dariusz Buraczewski, Gerold Alsmeyer","submitted_at":"2016-12-07T08:40:51Z","abstract_excerpt":"Given a sequence $(M_{k}, Q_{k})_{k\\ge 1}$ of independent, identically distributed ran\\-dom vectors with nonnegative components, we consider the recursive Markov chain $(X_{n})_{n\\ge 0}$, defined by the random difference equation $X_{n}=M_{n}X_{n-1}+Q_{n}$ for $n\\ge 1$, where $X_{0}$ is independent of $(M_{k}, Q_{k})_{k\\ge 1}$. Criteria for the null-recurrence/transience are provided in the situation where $(X_{n})_{n\\ge 0}$ is contractive in the sense that $M_{1}\\cdot\\ldots\\cdot M_{n}\\to 0$ a.s., yet occasional large values of the $Q_{n}$ overcompensate the contractive behavior so that positi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02148","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"}