{"paper":{"title":"Conditioned local limit theorems for random walks defined on finite Markov chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Emile Le Page, Ion Grama, Ronan Lauvergnat","submitted_at":"2017-07-19T14:48:42Z","abstract_excerpt":"Let $(X_n)_{n\\geq 0}$ be a Markov chain with values in a finite state space $\\mathbb X$ starting at $X_0=x \\in \\mathbb X$ and let $f$ be a real function defined on $\\mathbb X$. Set $S_n=\\sum_{k=1}^{n} f(X_k)$, $n\\geqslant 1$. For any $y \\in \\mathbb R$ denote by $\\tau_y$ the first time when $y+S_n$ becomes non-positive. We study the asymptotic behaviour of the probability $\\mathbb P_x \\left( y+S_{n} \\in [z,z+a] \\,,\\, \\tau_y > n \\right)$ as $n\\to+\\infty.$ We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.06129","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"}