{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:ALFFHHDAXSFLJIRIKDJ5RF5VV5","short_pith_number":"pith:ALFFHHDA","canonical_record":{"source":{"id":"2008.04389","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-08-10T19:53:57Z","cross_cats_sorted":[],"title_canon_sha256":"7b48a8a74108ce3e4f35e6e2e6bc2e670b833c2721a54e379120ea43ccd2a68e","abstract_canon_sha256":"b090a3825feaff08734b97b3ad0f2afc04d6f1489048a20b89e6340a850e5778"},"schema_version":"1.0"},"canonical_sha256":"02ca539c60bc8ab4a22850d3d897b5af558b5504cb003377798edbc21be3c9e9","source":{"kind":"arxiv","id":"2008.04389","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2008.04389","created_at":"2026-07-05T01:26:24Z"},{"alias_kind":"arxiv_version","alias_value":"2008.04389v1","created_at":"2026-07-05T01:26:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2008.04389","created_at":"2026-07-05T01:26:24Z"},{"alias_kind":"pith_short_12","alias_value":"ALFFHHDAXSFL","created_at":"2026-07-05T01:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"ALFFHHDAXSFLJIRI","created_at":"2026-07-05T01:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"ALFFHHDA","created_at":"2026-07-05T01:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:ALFFHHDAXSFLJIRIKDJ5RF5VV5","target":"record","payload":{"canonical_record":{"source":{"id":"2008.04389","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-08-10T19:53:57Z","cross_cats_sorted":[],"title_canon_sha256":"7b48a8a74108ce3e4f35e6e2e6bc2e670b833c2721a54e379120ea43ccd2a68e","abstract_canon_sha256":"b090a3825feaff08734b97b3ad0f2afc04d6f1489048a20b89e6340a850e5778"},"schema_version":"1.0"},"canonical_sha256":"02ca539c60bc8ab4a22850d3d897b5af558b5504cb003377798edbc21be3c9e9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:26:24.599280Z","signature_b64":"8Glwx0HpdCXTr8r5WcEgvAluW+1z6v3DJ51fQvSZy6iXwKyALD+UqiO4YDAs2EBNBzmTsP8USptoHpsnr2IqBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02ca539c60bc8ab4a22850d3d897b5af558b5504cb003377798edbc21be3c9e9","last_reissued_at":"2026-07-05T01:26:24.598947Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:26:24.598947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2008.04389","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":"2026-07-05T01:26:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BTVFdhrunjSSD8bjrgZP9sPtJG7UbI4l8FmfJKlKBr+UbtLi0Wteyc/e+YJfMT2rE3HmE5aAYKAO/v8lWdElDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T16:58:02.420154Z"},"content_sha256":"3c7d41b137af628f319a6d35341937afbc8f8a92d564a586d9b18fd51ed9557d","schema_version":"1.0","event_id":"sha256:3c7d41b137af628f319a6d35341937afbc8f8a92d564a586d9b18fd51ed9557d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:ALFFHHDAXSFLJIRIKDJ5RF5VV5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Central Limit Theorem in High Dimensions : The Optimal Bound on Dimension Growth Rate","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Debraj Das, Soumendra Lahiri","submitted_at":"2020-08-10T19:53:57Z","abstract_excerpt":"In this article, we try to give an answer to the simple question: ``\\textit{What is the critical growth rate of the dimension $p$ as a function of the sample size $n$ for which the Central Limit Theorem holds uniformly over the collection of $p$-dimensional hyper-rectangles ?''}. Specifically, we are interested in the normal approximation of suitably scaled versions of the sum $\\sum_{i=1}^{n}X_i$ in $\\mathcal{R}^p$ uniformly over the class of hyper-rectangles $\\mathcal{A}^{re}=\\{\\prod_{j=1}^{p}[a_j,b_j]\\cap\\mathcal{R}:-\\infty\\leq a_j\\leq b_j \\leq \\infty, j=1,\\ldots,p\\}$, where $X_1,\\dots,X_n$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2008.04389","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2008.04389/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T01:26:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/fBLo6TGCyPJSS87/nweTgaTzBBZP1cBJm/khJ7M29ow9RMfys8lUqXwFhPiWj27G25+PwrK1JsDKCJbZFl5Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T16:58:02.421299Z"},"content_sha256":"6ca1cacf87445c8d939b775b99915ffc5d922cf71679dca6cc83062d322140ea","schema_version":"1.0","event_id":"sha256:6ca1cacf87445c8d939b775b99915ffc5d922cf71679dca6cc83062d322140ea"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ALFFHHDAXSFLJIRIKDJ5RF5VV5/bundle.json","state_url":"https://pith.science/pith/ALFFHHDAXSFLJIRIKDJ5RF5VV5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ALFFHHDAXSFLJIRIKDJ5RF5VV5/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":"2026-08-14T16:58:02Z","links":{"resolver":"https://pith.science/pith/ALFFHHDAXSFLJIRIKDJ5RF5VV5","bundle":"https://pith.science/pith/ALFFHHDAXSFLJIRIKDJ5RF5VV5/bundle.json","state":"https://pith.science/pith/ALFFHHDAXSFLJIRIKDJ5RF5VV5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ALFFHHDAXSFLJIRIKDJ5RF5VV5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:ALFFHHDAXSFLJIRIKDJ5RF5VV5","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":"b090a3825feaff08734b97b3ad0f2afc04d6f1489048a20b89e6340a850e5778","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-08-10T19:53:57Z","title_canon_sha256":"7b48a8a74108ce3e4f35e6e2e6bc2e670b833c2721a54e379120ea43ccd2a68e"},"schema_version":"1.0","source":{"id":"2008.04389","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2008.04389","created_at":"2026-07-05T01:26:24Z"},{"alias_kind":"arxiv_version","alias_value":"2008.04389v1","created_at":"2026-07-05T01:26:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2008.04389","created_at":"2026-07-05T01:26:24Z"},{"alias_kind":"pith_short_12","alias_value":"ALFFHHDAXSFL","created_at":"2026-07-05T01:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"ALFFHHDAXSFLJIRI","created_at":"2026-07-05T01:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"ALFFHHDA","created_at":"2026-07-05T01:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:6ca1cacf87445c8d939b775b99915ffc5d922cf71679dca6cc83062d322140ea","target":"graph","created_at":"2026-07-05T01:26:24Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2008.04389/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this article, we try to give an answer to the simple question: ``\\textit{What is the critical growth rate of the dimension $p$ as a function of the sample size $n$ for which the Central Limit Theorem holds uniformly over the collection of $p$-dimensional hyper-rectangles ?''}. Specifically, we are interested in the normal approximation of suitably scaled versions of the sum $\\sum_{i=1}^{n}X_i$ in $\\mathcal{R}^p$ uniformly over the class of hyper-rectangles $\\mathcal{A}^{re}=\\{\\prod_{j=1}^{p}[a_j,b_j]\\cap\\mathcal{R}:-\\infty\\leq a_j\\leq b_j \\leq \\infty, j=1,\\ldots,p\\}$, where $X_1,\\dots,X_n$ ","authors_text":"Debraj Das, Soumendra Lahiri","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-08-10T19:53:57Z","title":"Central Limit Theorem in High Dimensions : The Optimal Bound on Dimension Growth Rate"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2008.04389","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":"sha256:3c7d41b137af628f319a6d35341937afbc8f8a92d564a586d9b18fd51ed9557d","target":"record","created_at":"2026-07-05T01:26:24Z","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":"b090a3825feaff08734b97b3ad0f2afc04d6f1489048a20b89e6340a850e5778","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-08-10T19:53:57Z","title_canon_sha256":"7b48a8a74108ce3e4f35e6e2e6bc2e670b833c2721a54e379120ea43ccd2a68e"},"schema_version":"1.0","source":{"id":"2008.04389","kind":"arxiv","version":1}},"canonical_sha256":"02ca539c60bc8ab4a22850d3d897b5af558b5504cb003377798edbc21be3c9e9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"02ca539c60bc8ab4a22850d3d897b5af558b5504cb003377798edbc21be3c9e9","first_computed_at":"2026-07-05T01:26:24.598947Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:26:24.598947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8Glwx0HpdCXTr8r5WcEgvAluW+1z6v3DJ51fQvSZy6iXwKyALD+UqiO4YDAs2EBNBzmTsP8USptoHpsnr2IqBA==","signature_status":"signed_v1","signed_at":"2026-07-05T01:26:24.599280Z","signed_message":"canonical_sha256_bytes"},"source_id":"2008.04389","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3c7d41b137af628f319a6d35341937afbc8f8a92d564a586d9b18fd51ed9557d","sha256:6ca1cacf87445c8d939b775b99915ffc5d922cf71679dca6cc83062d322140ea"],"state_sha256":"1f81833dca5b1abc91847b975274d14bd7c259450c36980d59ffed3ebc122752"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iTZkAZadWwZ2rUK8cSNqJbxqtA2Noyln+mT2IOdWAjkzGOQP8g4Iyn6o0Sjmbmxu1WEF7c+ljy0mJC5jj4lXBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T16:58:02.429819Z","bundle_sha256":"b838af423db6c049616b5aeb6970ea972a04a6df198379d7f7fc5d1c7452b224"}}