{"paper":{"title":"Level-crossings of symmetric random walks and their application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Vyacheslav M. Abramov","submitted_at":"2010-04-11T22:27:34Z","abstract_excerpt":"Let $X_1$, $X_2$, $...$ be a sequence of independently and identically distributed random variables with $\\mathsf{E}X_1=0$, and let $S_0=0$ and $S_t=S_{t-1}+X_t$, $t=1,2,...$, be a random walk. Denote $\\tau={cases}\\inf\\{t>1: S_t\\leq0\\}, &\\text{if} \\ X_1>0, 1, &\\text{otherwise}. {cases}$ Let $\\alpha$ denote a positive number, and let $L_\\alpha$ denote the number of level-crossings from the below (or above) across the level $\\alpha$ during the interval $[0, \\tau]$. Under quite general assumption, an inequality for the expected number of level-crossings is established. Under some special assumpti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.1850","kind":"arxiv","version":8},"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"}