{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:AZIPCDPBJXE4HESKO6OYUIYTMK","short_pith_number":"pith:AZIPCDPB","schema_version":"1.0","canonical_sha256":"0650f10de14dc9c3924a779d8a231362a8cee2c05e738190e8c68c9d5d3db809","source":{"kind":"arxiv","id":"1201.1089","version":1},"attestation_state":"computed","paper":{"title":"Sharp maximal inequalities for the moments of martingales and non-negative submartingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Adam Os\\c{e}kowski","submitted_at":"2012-01-05T09:18:24Z","abstract_excerpt":"In the paper we study sharp maximal inequalities for martingales and non-negative submartingales: if $f$, $g$ are martingales satisfying \\[|\\mathrm{d}g_n|\\leq|\\mathrm{d}f_n|,\\qquad n=0,1,2,...,\\] almost surely, then \\[\\Bigl\\|\\sup_{n\\geq0}|g_n|\\Bigr\\|_p\\leq p\\|f\\|_p,\\qquad p\\geq2,\\] and the inequality is sharp. Furthermore, if $\\alpha\\in[0,1]$, $f$ is a non-negative submartingale and $g$ satisfies \\[|\\mathrm{d}g_n|\\leq|\\mathrm{d}f_n|\\quad and\\quad |\\mathbb{E}(\\mathrm{d}g_{n+1}|\\mathcal {F}_n)|\\leq\\alpha\\mathbb{E}(\\mathrm{d}f_{n+1}|\\mathcal{F}_n),\\qquad n=0,1,2,...,\\] almost surely, then \\[\\Bigl"},"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":"1201.1089","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2012-01-05T09:18:24Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"3b55567a24a988ba54c87bfd874d188564ae14686c26495398c8f20928c15c6a","abstract_canon_sha256":"8ba06e91300e4a40896da804ffbfb73cdd0e769fffd63fe24d617abee6befaac"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:06.210379Z","signature_b64":"VKxq2vUy5OrARQOU+JI4zeDcmM2lnwxHxjz4PubGaF9/DpBbmAz+fggJvBAjkubfyCOLSo3anS+Y14pZEpPaAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0650f10de14dc9c3924a779d8a231362a8cee2c05e738190e8c68c9d5d3db809","last_reissued_at":"2026-05-18T04:05:06.209585Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:06.209585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp maximal inequalities for the moments of martingales and non-negative submartingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Adam Os\\c{e}kowski","submitted_at":"2012-01-05T09:18:24Z","abstract_excerpt":"In the paper we study sharp maximal inequalities for martingales and non-negative submartingales: if $f$, $g$ are martingales satisfying \\[|\\mathrm{d}g_n|\\leq|\\mathrm{d}f_n|,\\qquad n=0,1,2,...,\\] almost surely, then \\[\\Bigl\\|\\sup_{n\\geq0}|g_n|\\Bigr\\|_p\\leq p\\|f\\|_p,\\qquad p\\geq2,\\] and the inequality is sharp. Furthermore, if $\\alpha\\in[0,1]$, $f$ is a non-negative submartingale and $g$ satisfies \\[|\\mathrm{d}g_n|\\leq|\\mathrm{d}f_n|\\quad and\\quad |\\mathbb{E}(\\mathrm{d}g_{n+1}|\\mathcal {F}_n)|\\leq\\alpha\\mathbb{E}(\\mathrm{d}f_{n+1}|\\mathcal{F}_n),\\qquad n=0,1,2,...,\\] almost surely, then \\[\\Bigl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.1089","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":"1201.1089","created_at":"2026-05-18T04:05:06.209654+00:00"},{"alias_kind":"arxiv_version","alias_value":"1201.1089v1","created_at":"2026-05-18T04:05:06.209654+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.1089","created_at":"2026-05-18T04:05:06.209654+00:00"},{"alias_kind":"pith_short_12","alias_value":"AZIPCDPBJXE4","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_16","alias_value":"AZIPCDPBJXE4HESK","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_8","alias_value":"AZIPCDPB","created_at":"2026-05-18T12:26:58.693483+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/AZIPCDPBJXE4HESKO6OYUIYTMK","json":"https://pith.science/pith/AZIPCDPBJXE4HESKO6OYUIYTMK.json","graph_json":"https://pith.science/api/pith-number/AZIPCDPBJXE4HESKO6OYUIYTMK/graph.json","events_json":"https://pith.science/api/pith-number/AZIPCDPBJXE4HESKO6OYUIYTMK/events.json","paper":"https://pith.science/paper/AZIPCDPB"},"agent_actions":{"view_html":"https://pith.science/pith/AZIPCDPBJXE4HESKO6OYUIYTMK","download_json":"https://pith.science/pith/AZIPCDPBJXE4HESKO6OYUIYTMK.json","view_paper":"https://pith.science/paper/AZIPCDPB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1201.1089&json=true","fetch_graph":"https://pith.science/api/pith-number/AZIPCDPBJXE4HESKO6OYUIYTMK/graph.json","fetch_events":"https://pith.science/api/pith-number/AZIPCDPBJXE4HESKO6OYUIYTMK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AZIPCDPBJXE4HESKO6OYUIYTMK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AZIPCDPBJXE4HESKO6OYUIYTMK/action/storage_attestation","attest_author":"https://pith.science/pith/AZIPCDPBJXE4HESKO6OYUIYTMK/action/author_attestation","sign_citation":"https://pith.science/pith/AZIPCDPBJXE4HESKO6OYUIYTMK/action/citation_signature","submit_replication":"https://pith.science/pith/AZIPCDPBJXE4HESKO6OYUIYTMK/action/replication_record"}},"created_at":"2026-05-18T04:05:06.209654+00:00","updated_at":"2026-05-18T04:05:06.209654+00:00"}