{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:X5HO4JUUGXOFDVF3SSXVZCNP5I","short_pith_number":"pith:X5HO4JUU","canonical_record":{"source":{"id":"1509.02231","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-09-08T00:11:44Z","cross_cats_sorted":[],"title_canon_sha256":"7456a3e7f1ab906f30aa7c742238f7a33cc21f0d3f1a1ccb5462143eb31e97cb","abstract_canon_sha256":"8d163edee6e9432fcaafe983818f8816dae086cb23e50e0e43bcbdcbad0577b1"},"schema_version":"1.0"},"canonical_sha256":"bf4eee269435dc51d4bb94af5c89afea2875687b474294442206c37c1e0dd080","source":{"kind":"arxiv","id":"1509.02231","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.02231","created_at":"2026-05-18T00:21:00Z"},{"alias_kind":"arxiv_version","alias_value":"1509.02231v1","created_at":"2026-05-18T00:21:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.02231","created_at":"2026-05-18T00:21:00Z"},{"alias_kind":"pith_short_12","alias_value":"X5HO4JUUGXOF","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"X5HO4JUUGXOFDVF3","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"X5HO4JUU","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:X5HO4JUUGXOFDVF3SSXVZCNP5I","target":"record","payload":{"canonical_record":{"source":{"id":"1509.02231","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-09-08T00:11:44Z","cross_cats_sorted":[],"title_canon_sha256":"7456a3e7f1ab906f30aa7c742238f7a33cc21f0d3f1a1ccb5462143eb31e97cb","abstract_canon_sha256":"8d163edee6e9432fcaafe983818f8816dae086cb23e50e0e43bcbdcbad0577b1"},"schema_version":"1.0"},"canonical_sha256":"bf4eee269435dc51d4bb94af5c89afea2875687b474294442206c37c1e0dd080","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:00.252305Z","signature_b64":"KSdTOVBfjYHeIuVBYT/QB8KVhpsuGTO65K/9BbDA6DooyrTv6XbHqBoiZ7dm3koB+IQx2DC8/rO0Qx5bVOzuAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf4eee269435dc51d4bb94af5c89afea2875687b474294442206c37c1e0dd080","last_reissued_at":"2026-05-18T00:21:00.251841Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:00.251841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.02231","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":"2026-05-18T00:21:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1T+D4HZlUygBCjfBbaQlQVPEfWsec31iyHmWzUVExbRNmp0Y5zubMa/bD/ERoHqQWoLJ/mCfOQbcVo9+srwbCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T18:25:38.425182Z"},"content_sha256":"143412994ee1f6bef53378a55760ec3df55adbe1bcf99f6c2b80c10797bfb7ca","schema_version":"1.0","event_id":"sha256:143412994ee1f6bef53378a55760ec3df55adbe1bcf99f6c2b80c10797bfb7ca"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:X5HO4JUUGXOFDVF3SSXVZCNP5I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Djalil Chafa\\\"i, Konstantin Tikhomirov","submitted_at":"2015-09-08T00:11:44Z","abstract_excerpt":"Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko--Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.02231","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":"2026-05-18T00:21:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JL3A7/ARWc0dHxq0WJ/XyIA5MPrPnyQi441OX3gzxmBbgO4OSGTVEvZFOEoUuXWfQg9N/iMy/9igxu7PlVXFDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T18:25:38.425544Z"},"content_sha256":"9b8df3337b896ae386434104c09134a4ce36a18dd081075fd351bf9ff389eb03","schema_version":"1.0","event_id":"sha256:9b8df3337b896ae386434104c09134a4ce36a18dd081075fd351bf9ff389eb03"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X5HO4JUUGXOFDVF3SSXVZCNP5I/bundle.json","state_url":"https://pith.science/pith/X5HO4JUUGXOFDVF3SSXVZCNP5I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X5HO4JUUGXOFDVF3SSXVZCNP5I/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":"2026-06-02T18:25:38Z","links":{"resolver":"https://pith.science/pith/X5HO4JUUGXOFDVF3SSXVZCNP5I","bundle":"https://pith.science/pith/X5HO4JUUGXOFDVF3SSXVZCNP5I/bundle.json","state":"https://pith.science/pith/X5HO4JUUGXOFDVF3SSXVZCNP5I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X5HO4JUUGXOFDVF3SSXVZCNP5I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:X5HO4JUUGXOFDVF3SSXVZCNP5I","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":"8d163edee6e9432fcaafe983818f8816dae086cb23e50e0e43bcbdcbad0577b1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-09-08T00:11:44Z","title_canon_sha256":"7456a3e7f1ab906f30aa7c742238f7a33cc21f0d3f1a1ccb5462143eb31e97cb"},"schema_version":"1.0","source":{"id":"1509.02231","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.02231","created_at":"2026-05-18T00:21:00Z"},{"alias_kind":"arxiv_version","alias_value":"1509.02231v1","created_at":"2026-05-18T00:21:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.02231","created_at":"2026-05-18T00:21:00Z"},{"alias_kind":"pith_short_12","alias_value":"X5HO4JUUGXOF","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"X5HO4JUUGXOFDVF3","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"X5HO4JUU","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:9b8df3337b896ae386434104c09134a4ce36a18dd081075fd351bf9ff389eb03","target":"graph","created_at":"2026-05-18T00:21:00Z","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":"Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko--Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refi","authors_text":"Djalil Chafa\\\"i, Konstantin Tikhomirov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-09-08T00:11:44Z","title":"On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.02231","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":"sha256:143412994ee1f6bef53378a55760ec3df55adbe1bcf99f6c2b80c10797bfb7ca","target":"record","created_at":"2026-05-18T00:21:00Z","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":"8d163edee6e9432fcaafe983818f8816dae086cb23e50e0e43bcbdcbad0577b1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-09-08T00:11:44Z","title_canon_sha256":"7456a3e7f1ab906f30aa7c742238f7a33cc21f0d3f1a1ccb5462143eb31e97cb"},"schema_version":"1.0","source":{"id":"1509.02231","kind":"arxiv","version":1}},"canonical_sha256":"bf4eee269435dc51d4bb94af5c89afea2875687b474294442206c37c1e0dd080","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf4eee269435dc51d4bb94af5c89afea2875687b474294442206c37c1e0dd080","first_computed_at":"2026-05-18T00:21:00.251841Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:21:00.251841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KSdTOVBfjYHeIuVBYT/QB8KVhpsuGTO65K/9BbDA6DooyrTv6XbHqBoiZ7dm3koB+IQx2DC8/rO0Qx5bVOzuAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:21:00.252305Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.02231","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:143412994ee1f6bef53378a55760ec3df55adbe1bcf99f6c2b80c10797bfb7ca","sha256:9b8df3337b896ae386434104c09134a4ce36a18dd081075fd351bf9ff389eb03"],"state_sha256":"fad7e06e04d4ee4b9f3d9bc711f1aa330df0ee31631ae3117434ade1966fb691"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jDd1UKE56P9AYQT/PbvZzq0cshTx4W0plqD/6umH55akoBikRlecxull0rap7gDWy4Oe5HkR/2I1WmTszAWMDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T18:25:38.427428Z","bundle_sha256":"ddbf3f567b7adc4fab63e8a34c8d6387f14ee9e502e31899f45d93a3612d3f2b"}}