{"paper":{"title":"Renewal approximation for the absorption time of a decreasing Markov chain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander Marynych, Gerold Alsmeyer","submitted_at":"2015-09-05T14:48:19Z","abstract_excerpt":"We consider a Markov chain $(M_{n})_{n\\ge 0}$ on the set $\\mathbb{N}_{0}$ of nonnegative integers which is eventually decreasing, i.e. $\\mathbb{P}\\{M_{n+1}<M_{n}|M_{n}\\ge a\\}=1$ for some $a\\in\\mathbb{N}$ and all $n\\ge 0$. We are interested in the asymptotic behaviour of the law of the stopping time $T=T(a):=\\inf\\{k\\in\\mathbb{N}_{0}: M_{k}<a\\}$ under $\\mathbb{P}_{n}:=\\mathbb{P}(\\cdot|M_{0}=n)$ as $n\\to\\infty$. Assuming that the decrements of $(M_{n})_{n\\ge 0}$ given $M_{0}=n$ possess a kind of stationarity for large $n$, we derive sufficient conditions for the convergence in minimal $L^{p}$-dis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01704","kind":"arxiv","version":1},"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"}