{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:LX4H242ZG3CRE37MDL2V6LW3J5","short_pith_number":"pith:LX4H242Z","canonical_record":{"source":{"id":"1604.02964","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-04-11T14:04:39Z","cross_cats_sorted":[],"title_canon_sha256":"c5f8c6eefbed7a00708c1b56cd46c307d2c98ae13dfb5919e6babf4ef86a355d","abstract_canon_sha256":"67c279004abd0d427973eea83a667d2fa554dde410bab77ea7e4c8a08cd58b62"},"schema_version":"1.0"},"canonical_sha256":"5df87d735936c5126fec1af55f2edb4f4e956d5b665eccb93387517038610518","source":{"kind":"arxiv","id":"1604.02964","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.02964","created_at":"2026-05-17T23:51:34Z"},{"alias_kind":"arxiv_version","alias_value":"1604.02964v5","created_at":"2026-05-17T23:51:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.02964","created_at":"2026-05-17T23:51:34Z"},{"alias_kind":"pith_short_12","alias_value":"LX4H242ZG3CR","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"LX4H242ZG3CRE37M","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"LX4H242Z","created_at":"2026-05-18T12:30:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:LX4H242ZG3CRE37MDL2V6LW3J5","target":"record","payload":{"canonical_record":{"source":{"id":"1604.02964","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-04-11T14:04:39Z","cross_cats_sorted":[],"title_canon_sha256":"c5f8c6eefbed7a00708c1b56cd46c307d2c98ae13dfb5919e6babf4ef86a355d","abstract_canon_sha256":"67c279004abd0d427973eea83a667d2fa554dde410bab77ea7e4c8a08cd58b62"},"schema_version":"1.0"},"canonical_sha256":"5df87d735936c5126fec1af55f2edb4f4e956d5b665eccb93387517038610518","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:34.945010Z","signature_b64":"0XYtlu/fLfcF/1RsJM9q3anVBBuYeYuVHvBoUyQ8VTK4Zt8pYW80NtSqjiIzRJdzfhi+ghR4g/4cmoi0OCX3Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5df87d735936c5126fec1af55f2edb4f4e956d5b665eccb93387517038610518","last_reissued_at":"2026-05-17T23:51:34.944224Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:34.944224Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.02964","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:51:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dqzCb/9/YhrhJ84mNiWftrUN5GDFAcSLF4QP+5vrkeGSXZEoEPIEGP1uIbJ+52d87TdzSUeyPTLGEyNueOqqCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T23:21:54.611173Z"},"content_sha256":"129149847e592fe4cb73bb3ee082d1324bb55c1e65756003959dee6657c05d8c","schema_version":"1.0","event_id":"sha256:129149847e592fe4cb73bb3ee082d1324bb55c1e65756003959dee6657c05d8c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:LX4H242ZG3CRE37MDL2V6LW3J5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Central limit theorem analogues for multicolour urn models","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Noela M\\\"uller","submitted_at":"2016-04-11T14:04:39Z","abstract_excerpt":"The asymptotic behaviour of a generalised P\\'olya--Eggenberger urn is well--known to depend on the spectrum of its replacement matrix: If its dominant eigenvalue $r$ is simple and no other eigenvalue is `large' in the sense that its real part is greater than $r/2$, the normalized urn composition is asymptotically normally distributed. However, if there is more than one large eigenvalue, the first few random draws have a non--negligible effect on the evolution of the urn process and almost sure random tendencies of order larger than $\\sqrt{n}$ typically prevent a classical central limit theorem"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02964","kind":"arxiv","version":5},"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":"2026-05-17T23:51:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1vCKKvFKk3hds3A0jMEM3tkoVG8h9mVNo+lg+Zyaeeex2uRnrBNLix7oxqua/MLDRBZ8dprzMIhIJXZzvV33AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T23:21:54.611895Z"},"content_sha256":"1849427df0000bfeed9f0450361d1b0fb96fd6f9ecd309a70fd2ae16ee6f6666","schema_version":"1.0","event_id":"sha256:1849427df0000bfeed9f0450361d1b0fb96fd6f9ecd309a70fd2ae16ee6f6666"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LX4H242ZG3CRE37MDL2V6LW3J5/bundle.json","state_url":"https://pith.science/pith/LX4H242ZG3CRE37MDL2V6LW3J5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LX4H242ZG3CRE37MDL2V6LW3J5/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":"2026-06-06T23:21:54Z","links":{"resolver":"https://pith.science/pith/LX4H242ZG3CRE37MDL2V6LW3J5","bundle":"https://pith.science/pith/LX4H242ZG3CRE37MDL2V6LW3J5/bundle.json","state":"https://pith.science/pith/LX4H242ZG3CRE37MDL2V6LW3J5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LX4H242ZG3CRE37MDL2V6LW3J5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:LX4H242ZG3CRE37MDL2V6LW3J5","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":"67c279004abd0d427973eea83a667d2fa554dde410bab77ea7e4c8a08cd58b62","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-04-11T14:04:39Z","title_canon_sha256":"c5f8c6eefbed7a00708c1b56cd46c307d2c98ae13dfb5919e6babf4ef86a355d"},"schema_version":"1.0","source":{"id":"1604.02964","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.02964","created_at":"2026-05-17T23:51:34Z"},{"alias_kind":"arxiv_version","alias_value":"1604.02964v5","created_at":"2026-05-17T23:51:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.02964","created_at":"2026-05-17T23:51:34Z"},{"alias_kind":"pith_short_12","alias_value":"LX4H242ZG3CR","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"LX4H242ZG3CRE37M","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"LX4H242Z","created_at":"2026-05-18T12:30:29Z"}],"graph_snapshots":[{"event_id":"sha256:1849427df0000bfeed9f0450361d1b0fb96fd6f9ecd309a70fd2ae16ee6f6666","target":"graph","created_at":"2026-05-17T23:51:34Z","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":"The asymptotic behaviour of a generalised P\\'olya--Eggenberger urn is well--known to depend on the spectrum of its replacement matrix: If its dominant eigenvalue $r$ is simple and no other eigenvalue is `large' in the sense that its real part is greater than $r/2$, the normalized urn composition is asymptotically normally distributed. However, if there is more than one large eigenvalue, the first few random draws have a non--negligible effect on the evolution of the urn process and almost sure random tendencies of order larger than $\\sqrt{n}$ typically prevent a classical central limit theorem","authors_text":"Noela M\\\"uller","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-04-11T14:04:39Z","title":"Central limit theorem analogues for multicolour urn models"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02964","kind":"arxiv","version":5},"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":"sha256:129149847e592fe4cb73bb3ee082d1324bb55c1e65756003959dee6657c05d8c","target":"record","created_at":"2026-05-17T23:51:34Z","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":"67c279004abd0d427973eea83a667d2fa554dde410bab77ea7e4c8a08cd58b62","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-04-11T14:04:39Z","title_canon_sha256":"c5f8c6eefbed7a00708c1b56cd46c307d2c98ae13dfb5919e6babf4ef86a355d"},"schema_version":"1.0","source":{"id":"1604.02964","kind":"arxiv","version":5}},"canonical_sha256":"5df87d735936c5126fec1af55f2edb4f4e956d5b665eccb93387517038610518","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5df87d735936c5126fec1af55f2edb4f4e956d5b665eccb93387517038610518","first_computed_at":"2026-05-17T23:51:34.944224Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:34.944224Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0XYtlu/fLfcF/1RsJM9q3anVBBuYeYuVHvBoUyQ8VTK4Zt8pYW80NtSqjiIzRJdzfhi+ghR4g/4cmoi0OCX3Dg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:34.945010Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.02964","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:129149847e592fe4cb73bb3ee082d1324bb55c1e65756003959dee6657c05d8c","sha256:1849427df0000bfeed9f0450361d1b0fb96fd6f9ecd309a70fd2ae16ee6f6666"],"state_sha256":"c7329f09a9243d5aca1e992d7427a32876ff197e87847d770f8bfe1968b60103"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0x1GDyodpHMROn5jKswQlMweyZtbmfe7hNGs/6uGocwpK0JiNQ/Jhii5QDlDPd5OKtbGkDJJqNS360KxzwPjAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T23:21:54.615834Z","bundle_sha256":"9509f549452106f6c0fbbcac70a940c860e238360d61d422502e6972d6f4a9e5"}}