{"paper":{"title":"Small and Large Time Stability of the Time taken for a L\\'evy Process to Cross Curved Boundaries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Philip S. Griffin, Ross A. Maller","submitted_at":"2011-10-13T20:42:29Z","abstract_excerpt":"This paper is concerned with the small time behaviour of a L\\'{e}vy process $X$. In particular, we investigate the {\\it stabilities} of the times, $\\Tstarb(r)$ and $\\Tbarb(r)$, at which $X$, started with $X_0=0$, first leaves the space-time regions $\\{(t,y)\\in\\R^2: y\\le rt^b, t\\ge 0\\}$ (one-sided exit), or $\\{(t,y)\\in\\R^2: |y|\\le rt^b, t\\ge 0\\}$ (two-sided exit), $0\\le b<1$, as $r\\dto 0$. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.3064","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"}