{"paper":{"title":"Convergence of the all-time supremum of a L\\'evy process in the heavy-traffic regime","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Bert Zwart, Kamil Marcin Kosinski, Onno Boxma","submitted_at":"2010-07-01T12:55:13Z","abstract_excerpt":"In this paper we derive a technique of obtaining limit theorems for suprema of L\\'evy processes from their random walk counterparts. For each $a>0$, let $\\{Y^{(a)}_n:n\\ge 1\\}$ be a sequence of independent and identically distributed random variables and $\\{X^{(a)}_t:t\\ge 0\\}$ be a L\\'evy processes such that $X_1^{(a)}\\stackrel{d}{=} Y_1^{(a)}$, $\\mathbb E X_1^{(a)}<0$ and $\\mathbb E X_1^{(a)}\\uparrow0$ as $a\\downarrow0$. Let $S^{(a)}_n=\\sum_{k=1}^n Y^{(a)}_k$. Then, under some mild assumptions, $\\Delta(a)\\max_{n\\ge 0} S_n^{(a)}\\stackrel{d}{\\to} R\\iff\\Delta(a)\\sup_{t\\ge 0} X^{(a)}_t\\stackrel{d}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.0155","kind":"arxiv","version":3},"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"}