{"paper":{"title":"Martingale approach to subexponential asymptotics for random walks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Denis Denisov, Vitali Wachtel","submitted_at":"2011-11-29T13:37:39Z","abstract_excerpt":"Consider the random walk $S_n=\\xi_1+...+\\xi_n$ with independent and identically distributed increments and negative mean $\\mathbf E\\xi=-m<0$. Let $M=\\sup_{0\\le i} S_i$ be the supremum of the random walk. In this note we present derivation of asymptotics for $\\mathbf P(M>x), x\\to\\infty$ for long-tailed distributions. This derivation is based on the martingale arguments and does not require any prior knowledge of the theory of long-tailed distributions. In addition the same approach allows to obtain asymptotics for $\\mathbf P(M_\\tau>x)$, where $M_\\tau=\\max_{0\\le i<\\tau}S_i$ and $\\tau=\\min\\{n\\ge "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6810","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"}