{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:46XBVI2KUTAH3Q2QN7Z2P7ZSBH","short_pith_number":"pith:46XBVI2K","schema_version":"1.0","canonical_sha256":"e7ae1aa34aa4c07dc3506ff3a7ff3209c825c7c68ad3c9dbc352147bed41cba0","source":{"kind":"arxiv","id":"1501.04308","version":2},"attestation_state":"computed","paper":{"title":"Some Insights About the Small Ball Probability Factorization for Hilbert Random Elements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.AP","stat.ME","stat.ML","stat.TH"],"primary_cat":"math.PR","authors_text":"Aldo Goia, Enea Bongiorno","submitted_at":"2015-01-18T14:48:30Z","abstract_excerpt":"Asymptotic factorizations for the small-ball probability (SmBP) of a Hilbert valued random element $X$ are rigorously established and discussed. In particular, given the first $d$ principal components (PCs) and as the radius $\\varepsilon$ of the ball tends to zero, the SmBP is asymptotically proportional to (a) the joint density of the first $d$ PCs, (b) the volume of the $d$-dimensional ball with radius $\\varepsilon$, and (c) a correction factor weighting the use of a truncated version of the process expansion. Moreover, under suitable assumptions on the spectrum of the covariance operator of"},"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":"1501.04308","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-18T14:48:30Z","cross_cats_sorted":["math.ST","stat.AP","stat.ME","stat.ML","stat.TH"],"title_canon_sha256":"36ab0dd95810fb1cadad7a79405137b086ab9772cab2571ce5b68d139cea76b8","abstract_canon_sha256":"997f17236102b7a19db4c29eb57703c44a3a84b072272c1819d5524b127ecb22"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:10.106763Z","signature_b64":"hol36ll5AaU8Pwy1FYCQ2ei7rz0qzfXCPoWCVn7sO06nru0R8JCzKB62VOvgCNdWRoiW73p12vBzYzah1nmdCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e7ae1aa34aa4c07dc3506ff3a7ff3209c825c7c68ad3c9dbc352147bed41cba0","last_reissued_at":"2026-05-18T01:18:10.106341Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:10.106341Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Insights About the Small Ball Probability Factorization for Hilbert Random Elements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.AP","stat.ME","stat.ML","stat.TH"],"primary_cat":"math.PR","authors_text":"Aldo Goia, Enea Bongiorno","submitted_at":"2015-01-18T14:48:30Z","abstract_excerpt":"Asymptotic factorizations for the small-ball probability (SmBP) of a Hilbert valued random element $X$ are rigorously established and discussed. In particular, given the first $d$ principal components (PCs) and as the radius $\\varepsilon$ of the ball tends to zero, the SmBP is asymptotically proportional to (a) the joint density of the first $d$ PCs, (b) the volume of the $d$-dimensional ball with radius $\\varepsilon$, and (c) a correction factor weighting the use of a truncated version of the process expansion. Moreover, under suitable assumptions on the spectrum of the covariance operator of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.04308","kind":"arxiv","version":2},"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":"1501.04308","created_at":"2026-05-18T01:18:10.106415+00:00"},{"alias_kind":"arxiv_version","alias_value":"1501.04308v2","created_at":"2026-05-18T01:18:10.106415+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.04308","created_at":"2026-05-18T01:18:10.106415+00:00"},{"alias_kind":"pith_short_12","alias_value":"46XBVI2KUTAH","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"46XBVI2KUTAH3Q2Q","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"46XBVI2K","created_at":"2026-05-18T12:29:05.191682+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/46XBVI2KUTAH3Q2QN7Z2P7ZSBH","json":"https://pith.science/pith/46XBVI2KUTAH3Q2QN7Z2P7ZSBH.json","graph_json":"https://pith.science/api/pith-number/46XBVI2KUTAH3Q2QN7Z2P7ZSBH/graph.json","events_json":"https://pith.science/api/pith-number/46XBVI2KUTAH3Q2QN7Z2P7ZSBH/events.json","paper":"https://pith.science/paper/46XBVI2K"},"agent_actions":{"view_html":"https://pith.science/pith/46XBVI2KUTAH3Q2QN7Z2P7ZSBH","download_json":"https://pith.science/pith/46XBVI2KUTAH3Q2QN7Z2P7ZSBH.json","view_paper":"https://pith.science/paper/46XBVI2K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1501.04308&json=true","fetch_graph":"https://pith.science/api/pith-number/46XBVI2KUTAH3Q2QN7Z2P7ZSBH/graph.json","fetch_events":"https://pith.science/api/pith-number/46XBVI2KUTAH3Q2QN7Z2P7ZSBH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/46XBVI2KUTAH3Q2QN7Z2P7ZSBH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/46XBVI2KUTAH3Q2QN7Z2P7ZSBH/action/storage_attestation","attest_author":"https://pith.science/pith/46XBVI2KUTAH3Q2QN7Z2P7ZSBH/action/author_attestation","sign_citation":"https://pith.science/pith/46XBVI2KUTAH3Q2QN7Z2P7ZSBH/action/citation_signature","submit_replication":"https://pith.science/pith/46XBVI2KUTAH3Q2QN7Z2P7ZSBH/action/replication_record"}},"created_at":"2026-05-18T01:18:10.106415+00:00","updated_at":"2026-05-18T01:18:10.106415+00:00"}