{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:FM3MCTUGMGHQG2XC7G6TNH6PE2","short_pith_number":"pith:FM3MCTUG","canonical_record":{"source":{"id":"1903.09864","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-03-23T18:45:15Z","cross_cats_sorted":[],"title_canon_sha256":"73409305787aa36b913137ec219855442b5dfc063a2cfebb3bffdc1f36b521b6","abstract_canon_sha256":"1212063acad21bcfb50fc67db3e2cc8219d92faa73393c72f0f5071532ad200c"},"schema_version":"1.0"},"canonical_sha256":"2b36c14e86618f036ae2f9bd369fcf269c0d531895bc6316d722b4b982559ff1","source":{"kind":"arxiv","id":"1903.09864","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.09864","created_at":"2026-05-17T23:50:35Z"},{"alias_kind":"arxiv_version","alias_value":"1903.09864v1","created_at":"2026-05-17T23:50:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.09864","created_at":"2026-05-17T23:50:35Z"},{"alias_kind":"pith_short_12","alias_value":"FM3MCTUGMGHQ","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_16","alias_value":"FM3MCTUGMGHQG2XC","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_8","alias_value":"FM3MCTUG","created_at":"2026-05-18T12:33:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:FM3MCTUGMGHQG2XC7G6TNH6PE2","target":"record","payload":{"canonical_record":{"source":{"id":"1903.09864","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-03-23T18:45:15Z","cross_cats_sorted":[],"title_canon_sha256":"73409305787aa36b913137ec219855442b5dfc063a2cfebb3bffdc1f36b521b6","abstract_canon_sha256":"1212063acad21bcfb50fc67db3e2cc8219d92faa73393c72f0f5071532ad200c"},"schema_version":"1.0"},"canonical_sha256":"2b36c14e86618f036ae2f9bd369fcf269c0d531895bc6316d722b4b982559ff1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:35.277607Z","signature_b64":"O1AYG0+MquV2VhWKGH1wQm/NHMKZ3tRRhOc1YhmjAkY7kJ1+kBlNhFuEsFDXUaulqlu8jxycXwQSOdBGZKJaCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b36c14e86618f036ae2f9bd369fcf269c0d531895bc6316d722b4b982559ff1","last_reissued_at":"2026-05-17T23:50:35.276897Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:35.276897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1903.09864","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":"2026-05-17T23:50:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gRaqVKhiQP6Hlfzzjj6VdR2h7nvGXEVHje06VopM/dk+3bRdtBk+FDD8MzhLJsbzIbgc6vhDtPK9hc2Rbo0YCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T05:16:06.482362Z"},"content_sha256":"e81f58ef24f77a6160e99a5b0a8be8af6ed3b0b5210c85aeef41bddb14061e68","schema_version":"1.0","event_id":"sha256:e81f58ef24f77a6160e99a5b0a8be8af6ed3b0b5210c85aeef41bddb14061e68"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:FM3MCTUGMGHQG2XC7G6TNH6PE2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Uniform approximation in classical weak convergence theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hajo Holzmann, Viktor Bengs","submitted_at":"2019-03-23T18:45:15Z","abstract_excerpt":"A common statistical task lies in showing asymptotic normality of certain statistics. In many of these situations, classical textbook results on weak convergence theory suffice for the problem at hand. However, there are quite some scenarios where stronger results are needed in order to establish an asymptotic normal approximation uniformly over a family of probability measures. In this note we collect some results in this direction. We restrict ourselves to weak convergence in $\\mathbb R^d$ with continuous limit measures."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.09864","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":"2026-05-17T23:50:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qe1MaiSh8dREUWq+iIkBbuhKlwFfUER8RsYwF2+nVgcG5jnM2r+a5MflXWoppyxC+2t0/DRDXZRK9G85pge5Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T05:16:06.483018Z"},"content_sha256":"0b0f0048703947d787ffe9c35dd37a7b6dc6c4b661a750c61d8c65bfb771610f","schema_version":"1.0","event_id":"sha256:0b0f0048703947d787ffe9c35dd37a7b6dc6c4b661a750c61d8c65bfb771610f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FM3MCTUGMGHQG2XC7G6TNH6PE2/bundle.json","state_url":"https://pith.science/pith/FM3MCTUGMGHQG2XC7G6TNH6PE2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FM3MCTUGMGHQG2XC7G6TNH6PE2/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":"2026-06-07T05:16:06Z","links":{"resolver":"https://pith.science/pith/FM3MCTUGMGHQG2XC7G6TNH6PE2","bundle":"https://pith.science/pith/FM3MCTUGMGHQG2XC7G6TNH6PE2/bundle.json","state":"https://pith.science/pith/FM3MCTUGMGHQG2XC7G6TNH6PE2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FM3MCTUGMGHQG2XC7G6TNH6PE2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:FM3MCTUGMGHQG2XC7G6TNH6PE2","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":"1212063acad21bcfb50fc67db3e2cc8219d92faa73393c72f0f5071532ad200c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-03-23T18:45:15Z","title_canon_sha256":"73409305787aa36b913137ec219855442b5dfc063a2cfebb3bffdc1f36b521b6"},"schema_version":"1.0","source":{"id":"1903.09864","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.09864","created_at":"2026-05-17T23:50:35Z"},{"alias_kind":"arxiv_version","alias_value":"1903.09864v1","created_at":"2026-05-17T23:50:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.09864","created_at":"2026-05-17T23:50:35Z"},{"alias_kind":"pith_short_12","alias_value":"FM3MCTUGMGHQ","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_16","alias_value":"FM3MCTUGMGHQG2XC","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_8","alias_value":"FM3MCTUG","created_at":"2026-05-18T12:33:15Z"}],"graph_snapshots":[{"event_id":"sha256:0b0f0048703947d787ffe9c35dd37a7b6dc6c4b661a750c61d8c65bfb771610f","target":"graph","created_at":"2026-05-17T23:50:35Z","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":"A common statistical task lies in showing asymptotic normality of certain statistics. In many of these situations, classical textbook results on weak convergence theory suffice for the problem at hand. However, there are quite some scenarios where stronger results are needed in order to establish an asymptotic normal approximation uniformly over a family of probability measures. In this note we collect some results in this direction. We restrict ourselves to weak convergence in $\\mathbb R^d$ with continuous limit measures.","authors_text":"Hajo Holzmann, Viktor Bengs","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-03-23T18:45:15Z","title":"Uniform approximation in classical weak convergence theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.09864","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:e81f58ef24f77a6160e99a5b0a8be8af6ed3b0b5210c85aeef41bddb14061e68","target":"record","created_at":"2026-05-17T23:50:35Z","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":"1212063acad21bcfb50fc67db3e2cc8219d92faa73393c72f0f5071532ad200c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-03-23T18:45:15Z","title_canon_sha256":"73409305787aa36b913137ec219855442b5dfc063a2cfebb3bffdc1f36b521b6"},"schema_version":"1.0","source":{"id":"1903.09864","kind":"arxiv","version":1}},"canonical_sha256":"2b36c14e86618f036ae2f9bd369fcf269c0d531895bc6316d722b4b982559ff1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b36c14e86618f036ae2f9bd369fcf269c0d531895bc6316d722b4b982559ff1","first_computed_at":"2026-05-17T23:50:35.276897Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:50:35.276897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"O1AYG0+MquV2VhWKGH1wQm/NHMKZ3tRRhOc1YhmjAkY7kJ1+kBlNhFuEsFDXUaulqlu8jxycXwQSOdBGZKJaCQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:50:35.277607Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.09864","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e81f58ef24f77a6160e99a5b0a8be8af6ed3b0b5210c85aeef41bddb14061e68","sha256:0b0f0048703947d787ffe9c35dd37a7b6dc6c4b661a750c61d8c65bfb771610f"],"state_sha256":"17b3132a32d4a61d97becf1107a4e9f40a9d583c58529bc9bcd9c70aea01086b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GW6G1c9y+Iu4g+oJG/AAPKvYnWpme+f19EAPrwSR7Aksa/JPUdCFc3Er4QZ8YF8DA7XywRTFcWVjXCvB1AAhCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T05:16:06.486229Z","bundle_sha256":"fb52a45a9228c5ce5548831a6bf7ef19c4502683ff88f34e9072a70d27ec53dd"}}