{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:TEEPYQCBIYDUBY46NVCLYXBBTD","short_pith_number":"pith:TEEPYQCB","canonical_record":{"source":{"id":"1102.1842","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-09T11:46:54Z","cross_cats_sorted":[],"title_canon_sha256":"4500ee9764313fdb1322705ae6da79ba1b9fca6658b4d241fb5f488b1d717d75","abstract_canon_sha256":"8fbfd67614e3f5e6f9be4af9aa7d47460ab256ceb9ebe85dae149ba117ddc5e2"},"schema_version":"1.0"},"canonical_sha256":"9908fc4041460740e39e6d44bc5c2198d0ebd365419afe6e0bf24f6b80d84e49","source":{"kind":"arxiv","id":"1102.1842","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.1842","created_at":"2026-05-18T03:59:27Z"},{"alias_kind":"arxiv_version","alias_value":"1102.1842v6","created_at":"2026-05-18T03:59:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.1842","created_at":"2026-05-18T03:59:27Z"},{"alias_kind":"pith_short_12","alias_value":"TEEPYQCBIYDU","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"TEEPYQCBIYDUBY46","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"TEEPYQCB","created_at":"2026-05-18T12:26:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:TEEPYQCBIYDUBY46NVCLYXBBTD","target":"record","payload":{"canonical_record":{"source":{"id":"1102.1842","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-09T11:46:54Z","cross_cats_sorted":[],"title_canon_sha256":"4500ee9764313fdb1322705ae6da79ba1b9fca6658b4d241fb5f488b1d717d75","abstract_canon_sha256":"8fbfd67614e3f5e6f9be4af9aa7d47460ab256ceb9ebe85dae149ba117ddc5e2"},"schema_version":"1.0"},"canonical_sha256":"9908fc4041460740e39e6d44bc5c2198d0ebd365419afe6e0bf24f6b80d84e49","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:27.542488Z","signature_b64":"C60TDUegDKN1CmmWBctcwb2ibDvQToroPnCPUuFU/5Aoicg48VFRfbYY3803z2ZpTYw3qD2dft3OnkdN6jz4Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9908fc4041460740e39e6d44bc5c2198d0ebd365419afe6e0bf24f6b80d84e49","last_reissued_at":"2026-05-18T03:59:27.541331Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:27.541331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1102.1842","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:59:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kgeGEehi7I3mQ68mHSsHzIF+zdE0VmAZn2dSwHBqZ8Ah9rR1q37b73RT0HqKt49TbyWbxBZtyg1hGBEULuwEBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T04:11:25.480595Z"},"content_sha256":"6d0df3972eea07bf1e41bc657eb5d1c1d7a906deb001aa93af923f2e46a7aa28","schema_version":"1.0","event_id":"sha256:6d0df3972eea07bf1e41bc657eb5d1c1d7a906deb001aa93af923f2e46a7aa28"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:TEEPYQCBIYDUBY46NVCLYXBBTD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Central limit theorem for Markov processes with spectral gap in the Wasserstein metric","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Anna Walczuk, Tomasz Komorowski","submitted_at":"2011-02-09T11:46:54Z","abstract_excerpt":"Suppose that $\\{X_t,\\,t\\ge0\\}$ is a non-stationary Markov process, taking values in a Polish metric space $E$. We prove the law of large numbers and central limit theorem for an additive functional of the form $\\int_0^T\\psi(X_s)ds$, provided that the dual transition probability semigroup, defined on measures, is strongly contractive in an appropriate Wasserstein metric. Function $\\psi$ is assumed to be Lipschitz on $E$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.1842","kind":"arxiv","version":6},"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":"2026-05-18T03:59:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zfub/Ai/DJyhTTzYZ3myYr79f8LV8x1p2tuWZnsNyTWNf0kXbakPC1KMqmpz1kOFKWQB0sd3a8RoOhBrok6QDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T04:11:25.481044Z"},"content_sha256":"ecd6b58c4df30838b2d244a674102b402ea1a45b353a8e1af9af3692018d0fe3","schema_version":"1.0","event_id":"sha256:ecd6b58c4df30838b2d244a674102b402ea1a45b353a8e1af9af3692018d0fe3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TEEPYQCBIYDUBY46NVCLYXBBTD/bundle.json","state_url":"https://pith.science/pith/TEEPYQCBIYDUBY46NVCLYXBBTD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TEEPYQCBIYDUBY46NVCLYXBBTD/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":"2026-05-27T04:11:25Z","links":{"resolver":"https://pith.science/pith/TEEPYQCBIYDUBY46NVCLYXBBTD","bundle":"https://pith.science/pith/TEEPYQCBIYDUBY46NVCLYXBBTD/bundle.json","state":"https://pith.science/pith/TEEPYQCBIYDUBY46NVCLYXBBTD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TEEPYQCBIYDUBY46NVCLYXBBTD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:TEEPYQCBIYDUBY46NVCLYXBBTD","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":"8fbfd67614e3f5e6f9be4af9aa7d47460ab256ceb9ebe85dae149ba117ddc5e2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-09T11:46:54Z","title_canon_sha256":"4500ee9764313fdb1322705ae6da79ba1b9fca6658b4d241fb5f488b1d717d75"},"schema_version":"1.0","source":{"id":"1102.1842","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.1842","created_at":"2026-05-18T03:59:27Z"},{"alias_kind":"arxiv_version","alias_value":"1102.1842v6","created_at":"2026-05-18T03:59:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.1842","created_at":"2026-05-18T03:59:27Z"},{"alias_kind":"pith_short_12","alias_value":"TEEPYQCBIYDU","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"TEEPYQCBIYDUBY46","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"TEEPYQCB","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:ecd6b58c4df30838b2d244a674102b402ea1a45b353a8e1af9af3692018d0fe3","target":"graph","created_at":"2026-05-18T03:59:27Z","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":"Suppose that $\\{X_t,\\,t\\ge0\\}$ is a non-stationary Markov process, taking values in a Polish metric space $E$. We prove the law of large numbers and central limit theorem for an additive functional of the form $\\int_0^T\\psi(X_s)ds$, provided that the dual transition probability semigroup, defined on measures, is strongly contractive in an appropriate Wasserstein metric. Function $\\psi$ is assumed to be Lipschitz on $E$.","authors_text":"Anna Walczuk, Tomasz Komorowski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-09T11:46:54Z","title":"Central limit theorem for Markov processes with spectral gap in the Wasserstein metric"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.1842","kind":"arxiv","version":6},"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":"sha256:6d0df3972eea07bf1e41bc657eb5d1c1d7a906deb001aa93af923f2e46a7aa28","target":"record","created_at":"2026-05-18T03:59:27Z","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":"8fbfd67614e3f5e6f9be4af9aa7d47460ab256ceb9ebe85dae149ba117ddc5e2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-09T11:46:54Z","title_canon_sha256":"4500ee9764313fdb1322705ae6da79ba1b9fca6658b4d241fb5f488b1d717d75"},"schema_version":"1.0","source":{"id":"1102.1842","kind":"arxiv","version":6}},"canonical_sha256":"9908fc4041460740e39e6d44bc5c2198d0ebd365419afe6e0bf24f6b80d84e49","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9908fc4041460740e39e6d44bc5c2198d0ebd365419afe6e0bf24f6b80d84e49","first_computed_at":"2026-05-18T03:59:27.541331Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:59:27.541331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"C60TDUegDKN1CmmWBctcwb2ibDvQToroPnCPUuFU/5Aoicg48VFRfbYY3803z2ZpTYw3qD2dft3OnkdN6jz4Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:59:27.542488Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.1842","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d0df3972eea07bf1e41bc657eb5d1c1d7a906deb001aa93af923f2e46a7aa28","sha256:ecd6b58c4df30838b2d244a674102b402ea1a45b353a8e1af9af3692018d0fe3"],"state_sha256":"7ce92f1615aba6dc0f59c60dee7acf201909902c93bdea6f4e4e2460843f0ea5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NwSmN9HU+dAVkG8wNwyua4aP70CMcuQXYimfmKh7P3Cxotk2t9ZlkwvfLK65o0SW6YeWILQ/mAhluHLZ/EccAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T04:11:25.484412Z","bundle_sha256":"b9fab2279218cd8c5fcae1a074cc58740b712cf813d4106772ddbc61341bf867"}}