{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:PBNHSO4SM2E32NOI5Q75EYTCSV","short_pith_number":"pith:PBNHSO4S","schema_version":"1.0","canonical_sha256":"785a793b926689bd35c8ec3fd2626295630ccdf328bd6927a5ccf7fccacb2468","source":{"kind":"arxiv","id":"1307.6044","version":1},"attestation_state":"computed","paper":{"title":"Self-normalized Cram\\'{e}r type moderate deviations for the maximum of sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Qi-Man Shao, Qiying Wang, Weidong Liu","submitted_at":"2013-07-23T12:41:33Z","abstract_excerpt":"Let $X_1,X_2,...$ be independent random variables with zero means and finite variances, and let $S_n=\\sum_{i=1}^nX_i$ and $V^2_n=\\sum_{i=1}^nX^2_i$. A Cram\\'{e}r type moderate deviation for the maximum of the self-normalized sums $\\max_{1\\leq k\\leq n}S_k/V_n$ is obtained. In particular, for identically distributed $X_1,X_2,...,$ it is proved that $P(\\max_{1\\leq k\\leq n}S_k\\geq xV_n)/(1-\\Phi (x))\\rightarrow2$ uniformly for $0<x\\leq\\mathrm{o}(n^{1/6})$ under the optimal finite third moment of $X_1$."},"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":"1307.6044","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2013-07-23T12:41:33Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"963cca95132b294b82477530a28f73e62f024571798d8648e2433df3682ceaf7","abstract_canon_sha256":"b5dcb20014927f3036d68a153c10f652b7bcb298f62dfd119d6927de6e06f3cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:17:45.480330Z","signature_b64":"dXDw0HKKj+PaeBp0fBTAOokLUdVIYVWsXRztAksI8NaDS3JHGkQ+2Z2S0XCxwbGyHp8WRow7ENr4923A21hkDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"785a793b926689bd35c8ec3fd2626295630ccdf328bd6927a5ccf7fccacb2468","last_reissued_at":"2026-05-18T03:17:45.479782Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:17:45.479782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Self-normalized Cram\\'{e}r type moderate deviations for the maximum of sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Qi-Man Shao, Qiying Wang, Weidong Liu","submitted_at":"2013-07-23T12:41:33Z","abstract_excerpt":"Let $X_1,X_2,...$ be independent random variables with zero means and finite variances, and let $S_n=\\sum_{i=1}^nX_i$ and $V^2_n=\\sum_{i=1}^nX^2_i$. A Cram\\'{e}r type moderate deviation for the maximum of the self-normalized sums $\\max_{1\\leq k\\leq n}S_k/V_n$ is obtained. In particular, for identically distributed $X_1,X_2,...,$ it is proved that $P(\\max_{1\\leq k\\leq n}S_k\\geq xV_n)/(1-\\Phi (x))\\rightarrow2$ uniformly for $0<x\\leq\\mathrm{o}(n^{1/6})$ under the optimal finite third moment of $X_1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.6044","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":"1307.6044","created_at":"2026-05-18T03:17:45.479882+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.6044v1","created_at":"2026-05-18T03:17:45.479882+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.6044","created_at":"2026-05-18T03:17:45.479882+00:00"},{"alias_kind":"pith_short_12","alias_value":"PBNHSO4SM2E3","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_16","alias_value":"PBNHSO4SM2E32NOI","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_8","alias_value":"PBNHSO4S","created_at":"2026-05-18T12:27:54.935989+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/PBNHSO4SM2E32NOI5Q75EYTCSV","json":"https://pith.science/pith/PBNHSO4SM2E32NOI5Q75EYTCSV.json","graph_json":"https://pith.science/api/pith-number/PBNHSO4SM2E32NOI5Q75EYTCSV/graph.json","events_json":"https://pith.science/api/pith-number/PBNHSO4SM2E32NOI5Q75EYTCSV/events.json","paper":"https://pith.science/paper/PBNHSO4S"},"agent_actions":{"view_html":"https://pith.science/pith/PBNHSO4SM2E32NOI5Q75EYTCSV","download_json":"https://pith.science/pith/PBNHSO4SM2E32NOI5Q75EYTCSV.json","view_paper":"https://pith.science/paper/PBNHSO4S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.6044&json=true","fetch_graph":"https://pith.science/api/pith-number/PBNHSO4SM2E32NOI5Q75EYTCSV/graph.json","fetch_events":"https://pith.science/api/pith-number/PBNHSO4SM2E32NOI5Q75EYTCSV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PBNHSO4SM2E32NOI5Q75EYTCSV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PBNHSO4SM2E32NOI5Q75EYTCSV/action/storage_attestation","attest_author":"https://pith.science/pith/PBNHSO4SM2E32NOI5Q75EYTCSV/action/author_attestation","sign_citation":"https://pith.science/pith/PBNHSO4SM2E32NOI5Q75EYTCSV/action/citation_signature","submit_replication":"https://pith.science/pith/PBNHSO4SM2E32NOI5Q75EYTCSV/action/replication_record"}},"created_at":"2026-05-18T03:17:45.479882+00:00","updated_at":"2026-05-18T03:17:45.479882+00:00"}