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Assume that the random walk is oscillating and asymptotically stable, that is, there exists a sequence $\\{c_n,n\\geq1\\}$ such that $S_n/c_n$ converges to a stable law. In this paper we determine the tail behaviour of $T_g$ for all oscillating asymptotically stable walks and all boundary sequences satisfying $g_n=o(c_n)$. Furthermore, we prove that the rescaled random walk conditione"},"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":"1801.04136","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-01-12T11:34:49Z","cross_cats_sorted":[],"title_canon_sha256":"96cea2c189a3b431cc38698d9929c3914ace81d521884ccf101dc071ee7219d7","abstract_canon_sha256":"7b29c38f44f62ae493da444e5d8c5e6e2ee31d9d66049995098ef29fb729ea35"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:09.691488Z","signature_b64":"BwJDVXsTnwrDE5mKkoGskzaarFHxuWKZU51r9bKPqJsNzwubAcxXmtPjfTJXyP4S66QLAx0TYh8ilD/eLelbAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"639a8303bf19711e8c6d6a0deabc1ab7e442ad7f37270f07e3af0eacf4edb7c0","last_reissued_at":"2026-05-18T00:26:09.690899Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:09.690899Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"First-passage times over moving boundaries for asymptotically stable walks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander Sakhanenko, Denis Denisov, Vitali Wachtel","submitted_at":"2018-01-12T11:34:49Z","abstract_excerpt":"Let $\\{S_n, n\\geq1\\}$ be a random walk wih independent and identically distributed increments and let $\\{g_n,n\\geq1\\}$ be a sequence of real numbers. Let $T_g$ denote the first time when $S_n$ leaves $(g_n,\\infty)$. Assume that the random walk is oscillating and asymptotically stable, that is, there exists a sequence $\\{c_n,n\\geq1\\}$ such that $S_n/c_n$ converges to a stable law. In this paper we determine the tail behaviour of $T_g$ for all oscillating asymptotically stable walks and all boundary sequences satisfying $g_n=o(c_n)$. 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