{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2008:M5GTSZH3YKXBX5B5WPVXVSIJO2","short_pith_number":"pith:M5GTSZH3","schema_version":"1.0","canonical_sha256":"674d3964fbc2ae1bf43db3eb7ac90976af920d0c8425fd2768b79f6262808669","source":{"kind":"arxiv","id":"0810.1061","version":1},"attestation_state":"computed","paper":{"title":"A Strong Law of Large Numbers with Applications to Self-Similar Stable Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Aklilu Zeleke, Erkan Nane, Yimin Xiao","submitted_at":"2008-10-06T20:44:23Z","abstract_excerpt":"Let $p \\in (0, \\infty)$ be a constant and let $\\{\\xi_n\\} \\subset L^p(\\Omega, {\\mathcal F}, \\P)$ be a sequence of random variables. For any integers $m, n \\ge 0$, denote $S_{m, n} = \\sum_{k=m}^{m + n} \\xi_k$. It is proved that, if there exist a nondecreasing function $\\varphi: \\R_+\\to \\R_+$ (which satisfies a mild regularity condition) and an appropriately chosen integer $a\\ge 2$ such that $$ \\sum_{n=0}^\\infty \\sup_{k \\ge 0} \\E\\bigg|\\frac{S_{k, a^n}} {\\varphi(a^n)} \\bigg|^p < \\infty,$$ Then $$ \\lim_{n \\to \\infty} \\frac{S_{0, n}} {\\varphi(n)} = 0\\qquad \\hbox{a.s.} $$ This extends Theorem 1 in Le"},"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":"0810.1061","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-10-06T20:44:23Z","cross_cats_sorted":[],"title_canon_sha256":"c8d7860dabeec7c6d6937981012c3bc551bbc501bf6544704c4d70fa0cd4d762","abstract_canon_sha256":"ae59a6b400a62dc44f28d29bc0bce86d76b4476bb03a197a412cfa7b8471efb5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:56.136297Z","signature_b64":"fa30EGh5eNLGtlAYqO5gbvCH+QrFymJoMmfDcwCGt2kwA6M6P+S2xxVgzaZmClTB7qi1LPZuo2n60ZZ+Vx0WBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"674d3964fbc2ae1bf43db3eb7ac90976af920d0c8425fd2768b79f6262808669","last_reissued_at":"2026-05-18T04:32:56.135552Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:56.135552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Strong Law of Large Numbers with Applications to Self-Similar Stable Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Aklilu Zeleke, Erkan Nane, Yimin Xiao","submitted_at":"2008-10-06T20:44:23Z","abstract_excerpt":"Let $p \\in (0, \\infty)$ be a constant and let $\\{\\xi_n\\} \\subset L^p(\\Omega, {\\mathcal F}, \\P)$ be a sequence of random variables. For any integers $m, n \\ge 0$, denote $S_{m, n} = \\sum_{k=m}^{m + n} \\xi_k$. It is proved that, if there exist a nondecreasing function $\\varphi: \\R_+\\to \\R_+$ (which satisfies a mild regularity condition) and an appropriately chosen integer $a\\ge 2$ such that $$ \\sum_{n=0}^\\infty \\sup_{k \\ge 0} \\E\\bigg|\\frac{S_{k, a^n}} {\\varphi(a^n)} \\bigg|^p < \\infty,$$ Then $$ \\lim_{n \\to \\infty} \\frac{S_{0, n}} {\\varphi(n)} = 0\\qquad \\hbox{a.s.} $$ This extends Theorem 1 in Le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0810.1061","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0810.1061","created_at":"2026-05-18T04:32:56.135665+00:00"},{"alias_kind":"arxiv_version","alias_value":"0810.1061v1","created_at":"2026-05-18T04:32:56.135665+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0810.1061","created_at":"2026-05-18T04:32:56.135665+00:00"},{"alias_kind":"pith_short_12","alias_value":"M5GTSZH3YKXB","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_16","alias_value":"M5GTSZH3YKXBX5B5","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_8","alias_value":"M5GTSZH3","created_at":"2026-05-18T12:25:57.157939+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/M5GTSZH3YKXBX5B5WPVXVSIJO2","json":"https://pith.science/pith/M5GTSZH3YKXBX5B5WPVXVSIJO2.json","graph_json":"https://pith.science/api/pith-number/M5GTSZH3YKXBX5B5WPVXVSIJO2/graph.json","events_json":"https://pith.science/api/pith-number/M5GTSZH3YKXBX5B5WPVXVSIJO2/events.json","paper":"https://pith.science/paper/M5GTSZH3"},"agent_actions":{"view_html":"https://pith.science/pith/M5GTSZH3YKXBX5B5WPVXVSIJO2","download_json":"https://pith.science/pith/M5GTSZH3YKXBX5B5WPVXVSIJO2.json","view_paper":"https://pith.science/paper/M5GTSZH3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0810.1061&json=true","fetch_graph":"https://pith.science/api/pith-number/M5GTSZH3YKXBX5B5WPVXVSIJO2/graph.json","fetch_events":"https://pith.science/api/pith-number/M5GTSZH3YKXBX5B5WPVXVSIJO2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/M5GTSZH3YKXBX5B5WPVXVSIJO2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/M5GTSZH3YKXBX5B5WPVXVSIJO2/action/storage_attestation","attest_author":"https://pith.science/pith/M5GTSZH3YKXBX5B5WPVXVSIJO2/action/author_attestation","sign_citation":"https://pith.science/pith/M5GTSZH3YKXBX5B5WPVXVSIJO2/action/citation_signature","submit_replication":"https://pith.science/pith/M5GTSZH3YKXBX5B5WPVXVSIJO2/action/replication_record"}},"created_at":"2026-05-18T04:32:56.135665+00:00","updated_at":"2026-05-18T04:32:56.135665+00:00"}