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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1110.3064","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-10-13T20:42:29Z","cross_cats_sorted":[],"title_canon_sha256":"cf517635a640cf51ca85939d215f0578950099f655de6f415e4a830e228a5e5d","abstract_canon_sha256":"cb2f588d8d87c516bfc50a11c25b74c18308fbee7a8ac8df2d484bc590b55006"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:58.232252Z","signature_b64":"CrZLb0Nws8gjndGRWuFFGYLlAlMH6KJQF1GjTTRr+iehmQghjsGgBF7+qzty5QuZRY5UAxQ12O6J2qRADXZaAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0bff66eaf5bbda4d2aa124ef7c22814f92b447b581985315615193aa1bde198","last_reissued_at":"2026-05-18T04:10:58.231580Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:58.231580Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $r\\dto 0$. 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