{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:KNJFRTFWZLHNADR5G5RUSG4FBR","short_pith_number":"pith:KNJFRTFW","schema_version":"1.0","canonical_sha256":"535258ccb6caced00e3d3763491b850c7eca7cdbcb61345696780b86c443792f","source":{"kind":"arxiv","id":"1701.03099","version":1},"attestation_state":"computed","paper":{"title":"On the Azuma inequality in spaces of subgaussian of rank $p$ random variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Krzysztof Zajkowski","submitted_at":"2017-01-11T18:58:44Z","abstract_excerpt":"For $p > 1$ let a function $\\varphi_p(x) = x^2/2$ if $|x|\\le 1$ and $\\varphi_p(x) = 1/p|x|^p -1/p + 1/2$ if $|x| > 1$. For a random variable $\\xi$ let $\\tau_{\\varphi_p}(\\xi)$ denote $\\inf\\{c\\ge 0 :\\; \\forall_{\\lambda\\in\\mathbb{R}}\\; \\ln\\mathbb{E}\\exp(\\lambda\\xi)\\le\\varphi_p(c\\lambda)\\}$; $\\tau_{\\varphi_p}$ is a norm in a space $Sub_{\\varphi_p}(\\Omega) =\\{\\xi:\n  \\; \\tau_{\\varphi_p}(\\xi) <\\infty\\}$ of $\\varphi_p$-subgaussian random variables which we call {\\it subgaussian of rank $p$ random variables}. For $p = 2$ we have the classic subgaussian random variables. The Azuma inequality gives an es"},"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":"1701.03099","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-01-11T18:58:44Z","cross_cats_sorted":[],"title_canon_sha256":"55a365ef7dde8f7f0b14140bee7ec71922b7de8d88cb38549a445f218383689d","abstract_canon_sha256":"701f7381b1068143a4ede9233123f4a73c1d6cb13df3b95cbb6e63ba5a71bbfd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:59.331020Z","signature_b64":"k0IDTzWc+Q19IGa2mGbjsfgqznFvYaJqttgwyHWnpzwAujr84B9sGd2crDoi8Igze+S6HMeMl50JI7ZF11icBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"535258ccb6caced00e3d3763491b850c7eca7cdbcb61345696780b86c443792f","last_reissued_at":"2026-05-18T00:52:59.330639Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:59.330639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Azuma inequality in spaces of subgaussian of rank $p$ random variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Krzysztof Zajkowski","submitted_at":"2017-01-11T18:58:44Z","abstract_excerpt":"For $p > 1$ let a function $\\varphi_p(x) = x^2/2$ if $|x|\\le 1$ and $\\varphi_p(x) = 1/p|x|^p -1/p + 1/2$ if $|x| > 1$. For a random variable $\\xi$ let $\\tau_{\\varphi_p}(\\xi)$ denote $\\inf\\{c\\ge 0 :\\; \\forall_{\\lambda\\in\\mathbb{R}}\\; \\ln\\mathbb{E}\\exp(\\lambda\\xi)\\le\\varphi_p(c\\lambda)\\}$; $\\tau_{\\varphi_p}$ is a norm in a space $Sub_{\\varphi_p}(\\Omega) =\\{\\xi:\n  \\; \\tau_{\\varphi_p}(\\xi) <\\infty\\}$ of $\\varphi_p$-subgaussian random variables which we call {\\it subgaussian of rank $p$ random variables}. For $p = 2$ we have the classic subgaussian random variables. The Azuma inequality gives an es"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.03099","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":"1701.03099","created_at":"2026-05-18T00:52:59.330699+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.03099v1","created_at":"2026-05-18T00:52:59.330699+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.03099","created_at":"2026-05-18T00:52:59.330699+00:00"},{"alias_kind":"pith_short_12","alias_value":"KNJFRTFWZLHN","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_16","alias_value":"KNJFRTFWZLHNADR5","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_8","alias_value":"KNJFRTFW","created_at":"2026-05-18T12:31:24.725408+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/KNJFRTFWZLHNADR5G5RUSG4FBR","json":"https://pith.science/pith/KNJFRTFWZLHNADR5G5RUSG4FBR.json","graph_json":"https://pith.science/api/pith-number/KNJFRTFWZLHNADR5G5RUSG4FBR/graph.json","events_json":"https://pith.science/api/pith-number/KNJFRTFWZLHNADR5G5RUSG4FBR/events.json","paper":"https://pith.science/paper/KNJFRTFW"},"agent_actions":{"view_html":"https://pith.science/pith/KNJFRTFWZLHNADR5G5RUSG4FBR","download_json":"https://pith.science/pith/KNJFRTFWZLHNADR5G5RUSG4FBR.json","view_paper":"https://pith.science/paper/KNJFRTFW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.03099&json=true","fetch_graph":"https://pith.science/api/pith-number/KNJFRTFWZLHNADR5G5RUSG4FBR/graph.json","fetch_events":"https://pith.science/api/pith-number/KNJFRTFWZLHNADR5G5RUSG4FBR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KNJFRTFWZLHNADR5G5RUSG4FBR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KNJFRTFWZLHNADR5G5RUSG4FBR/action/storage_attestation","attest_author":"https://pith.science/pith/KNJFRTFWZLHNADR5G5RUSG4FBR/action/author_attestation","sign_citation":"https://pith.science/pith/KNJFRTFWZLHNADR5G5RUSG4FBR/action/citation_signature","submit_replication":"https://pith.science/pith/KNJFRTFWZLHNADR5G5RUSG4FBR/action/replication_record"}},"created_at":"2026-05-18T00:52:59.330699+00:00","updated_at":"2026-05-18T00:52:59.330699+00:00"}