{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:XEVU72OFS2E3YGIYQUGKDS2WAZ","short_pith_number":"pith:XEVU72OF","canonical_record":{"source":{"id":"1402.6173","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-02-25T14:09:10Z","cross_cats_sorted":[],"title_canon_sha256":"6c1d073147fae1c808be85103789816ce80c0f47c233458e02bf83e22d892c76","abstract_canon_sha256":"82b907a8cee65e3212e80e0fa60d1a450ab1a047db4dcacea8972e0599a84c38"},"schema_version":"1.0"},"canonical_sha256":"b92b4fe9c59689bc1918850ca1cb56064c92f0cc3b065a982ee33a40784a08b5","source":{"kind":"arxiv","id":"1402.6173","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.6173","created_at":"2026-05-18T02:57:51Z"},{"alias_kind":"arxiv_version","alias_value":"1402.6173v1","created_at":"2026-05-18T02:57:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.6173","created_at":"2026-05-18T02:57:51Z"},{"alias_kind":"pith_short_12","alias_value":"XEVU72OFS2E3","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"XEVU72OFS2E3YGIY","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"XEVU72OF","created_at":"2026-05-18T12:28:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:XEVU72OFS2E3YGIYQUGKDS2WAZ","target":"record","payload":{"canonical_record":{"source":{"id":"1402.6173","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-02-25T14:09:10Z","cross_cats_sorted":[],"title_canon_sha256":"6c1d073147fae1c808be85103789816ce80c0f47c233458e02bf83e22d892c76","abstract_canon_sha256":"82b907a8cee65e3212e80e0fa60d1a450ab1a047db4dcacea8972e0599a84c38"},"schema_version":"1.0"},"canonical_sha256":"b92b4fe9c59689bc1918850ca1cb56064c92f0cc3b065a982ee33a40784a08b5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:51.444969Z","signature_b64":"oIRZom+/Lb4eFTFK22aiFCRdd3Q1GF0UgyviKMRC3jYLUI6QI7d9d6as0Z0hRZ9+wkuW5t0ojLVYLmj3Xcv5BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b92b4fe9c59689bc1918850ca1cb56064c92f0cc3b065a982ee33a40784a08b5","last_reissued_at":"2026-05-18T02:57:51.444441Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:51.444441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1402.6173","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:57:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3WvfEtV//GqkkaXd09qF3rysKaPEeH+j+uoOkZ87MIiOYeqYq3vPzcBOVbjQ98Uw8OfKPn4u7jzMO6asAMzMCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T19:43:20.162460Z"},"content_sha256":"13b6265e5deb435243f3ce4b957afd8efad40a0eeef518076a4593f676ebe59f","schema_version":"1.0","event_id":"sha256:13b6265e5deb435243f3ce4b957afd8efad40a0eeef518076a4593f676ebe59f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:XEVU72OFS2E3YGIYQUGKDS2WAZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Qi-Man Shao, Wen-Xin Zhou","submitted_at":"2014-02-25T14:09:10Z","abstract_excerpt":"Let $\\mathbf {x}_1,\\ldots,\\mathbf {x}_n$ be a random sample from a $p$-dimensional population distribution, where $p=p_n\\to\\infty$ and $\\log p=o(n^{\\beta})$ for some $0<\\beta\\leq1$, and let $L_n$ be the coherence of the sample correlation matrix. In this paper it is proved that $\\sqrt{n/\\log p}L_n\\to2$ in probability if and only if $Ee^{t_0|x_{11}|^{\\alpha}}<\\infty$ for some $t_0>0$, where $\\alpha$ satisfies $\\beta=\\alpha/(4-\\alpha)$. Asymptotic distributions of $L_n$ are also proved under the same sufficient condition. Similar results remain valid for $m$-coherence when the variables of the p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.6173","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:57:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k4k+bZP6dpYsjvD54t5HFR5iXuSIarzBKonA6Bmu1B9/vKL3NEPb7xlz7RXkBlidNB8alVZTKB+5YPmT7GcTAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T19:43:20.163253Z"},"content_sha256":"ae92d8f207024d49ec0a7ccfb87415fc59ad600bd405b344ad859dea0f2b4985","schema_version":"1.0","event_id":"sha256:ae92d8f207024d49ec0a7ccfb87415fc59ad600bd405b344ad859dea0f2b4985"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XEVU72OFS2E3YGIYQUGKDS2WAZ/bundle.json","state_url":"https://pith.science/pith/XEVU72OFS2E3YGIYQUGKDS2WAZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XEVU72OFS2E3YGIYQUGKDS2WAZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T19:43:20Z","links":{"resolver":"https://pith.science/pith/XEVU72OFS2E3YGIYQUGKDS2WAZ","bundle":"https://pith.science/pith/XEVU72OFS2E3YGIYQUGKDS2WAZ/bundle.json","state":"https://pith.science/pith/XEVU72OFS2E3YGIYQUGKDS2WAZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XEVU72OFS2E3YGIYQUGKDS2WAZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:XEVU72OFS2E3YGIYQUGKDS2WAZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"82b907a8cee65e3212e80e0fa60d1a450ab1a047db4dcacea8972e0599a84c38","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-02-25T14:09:10Z","title_canon_sha256":"6c1d073147fae1c808be85103789816ce80c0f47c233458e02bf83e22d892c76"},"schema_version":"1.0","source":{"id":"1402.6173","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.6173","created_at":"2026-05-18T02:57:51Z"},{"alias_kind":"arxiv_version","alias_value":"1402.6173v1","created_at":"2026-05-18T02:57:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.6173","created_at":"2026-05-18T02:57:51Z"},{"alias_kind":"pith_short_12","alias_value":"XEVU72OFS2E3","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"XEVU72OFS2E3YGIY","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"XEVU72OF","created_at":"2026-05-18T12:28:57Z"}],"graph_snapshots":[{"event_id":"sha256:ae92d8f207024d49ec0a7ccfb87415fc59ad600bd405b344ad859dea0f2b4985","target":"graph","created_at":"2026-05-18T02:57:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $\\mathbf {x}_1,\\ldots,\\mathbf {x}_n$ be a random sample from a $p$-dimensional population distribution, where $p=p_n\\to\\infty$ and $\\log p=o(n^{\\beta})$ for some $0<\\beta\\leq1$, and let $L_n$ be the coherence of the sample correlation matrix. In this paper it is proved that $\\sqrt{n/\\log p}L_n\\to2$ in probability if and only if $Ee^{t_0|x_{11}|^{\\alpha}}<\\infty$ for some $t_0>0$, where $\\alpha$ satisfies $\\beta=\\alpha/(4-\\alpha)$. Asymptotic distributions of $L_n$ are also proved under the same sufficient condition. Similar results remain valid for $m$-coherence when the variables of the p","authors_text":"Qi-Man Shao, Wen-Xin Zhou","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-02-25T14:09:10Z","title":"Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.6173","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:13b6265e5deb435243f3ce4b957afd8efad40a0eeef518076a4593f676ebe59f","target":"record","created_at":"2026-05-18T02:57:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"82b907a8cee65e3212e80e0fa60d1a450ab1a047db4dcacea8972e0599a84c38","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-02-25T14:09:10Z","title_canon_sha256":"6c1d073147fae1c808be85103789816ce80c0f47c233458e02bf83e22d892c76"},"schema_version":"1.0","source":{"id":"1402.6173","kind":"arxiv","version":1}},"canonical_sha256":"b92b4fe9c59689bc1918850ca1cb56064c92f0cc3b065a982ee33a40784a08b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b92b4fe9c59689bc1918850ca1cb56064c92f0cc3b065a982ee33a40784a08b5","first_computed_at":"2026-05-18T02:57:51.444441Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:57:51.444441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oIRZom+/Lb4eFTFK22aiFCRdd3Q1GF0UgyviKMRC3jYLUI6QI7d9d6as0Z0hRZ9+wkuW5t0ojLVYLmj3Xcv5BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:57:51.444969Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.6173","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:13b6265e5deb435243f3ce4b957afd8efad40a0eeef518076a4593f676ebe59f","sha256:ae92d8f207024d49ec0a7ccfb87415fc59ad600bd405b344ad859dea0f2b4985"],"state_sha256":"a31119405fbe3350090248c730f6d3962478a61abd115a24722f38e29a45a66e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r2n+TNmT4qqc5fjFO/6TIDIOyhIJtuRa6kwam6Ox7xQXxOlzw9NDD6VbC9aehspgLl0ljvqyrF7MWNJnyJvyCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T19:43:20.167058Z","bundle_sha256":"f61016dc395094f56d42ff656c89a233840ce50ed7838e45612804baaf92b5eb"}}