{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:LQAGRCN5OSK5ZHTXO7WGPNST2D","short_pith_number":"pith:LQAGRCN5","canonical_record":{"source":{"id":"1103.6266","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-31T18:45:03Z","cross_cats_sorted":[],"title_canon_sha256":"2d59bcf24bef7c3ca051e40b2398021f64f853925148476a6c59d319a6ab38c6","abstract_canon_sha256":"b7b2fe56387d02c3db0176c1094e33103c22dfa65ea615804904ebe810b8b0ff"},"schema_version":"1.0"},"canonical_sha256":"5c006889bd7495dc9e7777ec67b653d0d947f00ed7abea6d2b9a8b63fb023bae","source":{"kind":"arxiv","id":"1103.6266","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.6266","created_at":"2026-05-18T04:23:02Z"},{"alias_kind":"arxiv_version","alias_value":"1103.6266v2","created_at":"2026-05-18T04:23:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.6266","created_at":"2026-05-18T04:23:02Z"},{"alias_kind":"pith_short_12","alias_value":"LQAGRCN5OSK5","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"LQAGRCN5OSK5ZHTX","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"LQAGRCN5","created_at":"2026-05-18T12:26:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:LQAGRCN5OSK5ZHTXO7WGPNST2D","target":"record","payload":{"canonical_record":{"source":{"id":"1103.6266","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-31T18:45:03Z","cross_cats_sorted":[],"title_canon_sha256":"2d59bcf24bef7c3ca051e40b2398021f64f853925148476a6c59d319a6ab38c6","abstract_canon_sha256":"b7b2fe56387d02c3db0176c1094e33103c22dfa65ea615804904ebe810b8b0ff"},"schema_version":"1.0"},"canonical_sha256":"5c006889bd7495dc9e7777ec67b653d0d947f00ed7abea6d2b9a8b63fb023bae","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:02.176186Z","signature_b64":"VoFMXPC72tIUMdPac8/QpAqBZ4Rrb/QjR9bSNsYMnwyhVNWdkD8JO6tlQn92d1jHGI++1J0qB5abuqPJR3sfDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c006889bd7495dc9e7777ec67b653d0d947f00ed7abea6d2b9a8b63fb023bae","last_reissued_at":"2026-05-18T04:23:02.175282Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:02.175282Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.6266","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:23:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5P41/KTMxOARhYvuxD+LmmzRx9483H1POZd5jIwKJagVxtXhJp9T7qGDQBN1KQ8MP3sMUCMV/Q9NAI1fDDfDAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T22:56:49.609133Z"},"content_sha256":"f9eac650e5f1f87bd49012333057db3524daa47f0d92cbb626043b48b9e609e3","schema_version":"1.0","event_id":"sha256:f9eac650e5f1f87bd49012333057db3524daa47f0d92cbb626043b48b9e609e3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:LQAGRCN5OSK5ZHTXO7WGPNST2D","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Almost Sure Invariance Principles via Martingale Approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Costel Peligrad, Florence Merlev\\`ede, Magda Peligrad","submitted_at":"2011-03-31T18:45:03Z","abstract_excerpt":"In this paper we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale with stationary differences. The results are exploited to further investigate the central limit theorem and its invariance principle started at a point, as well as the law of the iterated logarithm via almost sure approximation with a Brownian motion, improving the results available in the literature. The conditions are well suited for a variety of examples; "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.6266","kind":"arxiv","version":2},"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":"2026-05-18T04:23:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jyB/ZqRiFaHSt+0PDsrpeaIRT/V5eKnzkueu/sMBTyJkGhXHMLMxMMwEJ5+xn1E9PabZ6ihXL8V4YjbTpFtNCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T22:56:49.609534Z"},"content_sha256":"67df1e5e6e8b397e9bf97df37bad16e213b603b564e8ef567b4320ae9cc270d5","schema_version":"1.0","event_id":"sha256:67df1e5e6e8b397e9bf97df37bad16e213b603b564e8ef567b4320ae9cc270d5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LQAGRCN5OSK5ZHTXO7WGPNST2D/bundle.json","state_url":"https://pith.science/pith/LQAGRCN5OSK5ZHTXO7WGPNST2D/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LQAGRCN5OSK5ZHTXO7WGPNST2D/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":"2026-05-28T22:56:49Z","links":{"resolver":"https://pith.science/pith/LQAGRCN5OSK5ZHTXO7WGPNST2D","bundle":"https://pith.science/pith/LQAGRCN5OSK5ZHTXO7WGPNST2D/bundle.json","state":"https://pith.science/pith/LQAGRCN5OSK5ZHTXO7WGPNST2D/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LQAGRCN5OSK5ZHTXO7WGPNST2D/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:LQAGRCN5OSK5ZHTXO7WGPNST2D","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":"b7b2fe56387d02c3db0176c1094e33103c22dfa65ea615804904ebe810b8b0ff","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-31T18:45:03Z","title_canon_sha256":"2d59bcf24bef7c3ca051e40b2398021f64f853925148476a6c59d319a6ab38c6"},"schema_version":"1.0","source":{"id":"1103.6266","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.6266","created_at":"2026-05-18T04:23:02Z"},{"alias_kind":"arxiv_version","alias_value":"1103.6266v2","created_at":"2026-05-18T04:23:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.6266","created_at":"2026-05-18T04:23:02Z"},{"alias_kind":"pith_short_12","alias_value":"LQAGRCN5OSK5","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"LQAGRCN5OSK5ZHTX","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"LQAGRCN5","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:67df1e5e6e8b397e9bf97df37bad16e213b603b564e8ef567b4320ae9cc270d5","target":"graph","created_at":"2026-05-18T04:23:02Z","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":"In this paper we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale with stationary differences. The results are exploited to further investigate the central limit theorem and its invariance principle started at a point, as well as the law of the iterated logarithm via almost sure approximation with a Brownian motion, improving the results available in the literature. The conditions are well suited for a variety of examples; ","authors_text":"Costel Peligrad, Florence Merlev\\`ede, Magda Peligrad","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-31T18:45:03Z","title":"Almost Sure Invariance Principles via Martingale Approximation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.6266","kind":"arxiv","version":2},"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:f9eac650e5f1f87bd49012333057db3524daa47f0d92cbb626043b48b9e609e3","target":"record","created_at":"2026-05-18T04:23:02Z","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":"b7b2fe56387d02c3db0176c1094e33103c22dfa65ea615804904ebe810b8b0ff","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-31T18:45:03Z","title_canon_sha256":"2d59bcf24bef7c3ca051e40b2398021f64f853925148476a6c59d319a6ab38c6"},"schema_version":"1.0","source":{"id":"1103.6266","kind":"arxiv","version":2}},"canonical_sha256":"5c006889bd7495dc9e7777ec67b653d0d947f00ed7abea6d2b9a8b63fb023bae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5c006889bd7495dc9e7777ec67b653d0d947f00ed7abea6d2b9a8b63fb023bae","first_computed_at":"2026-05-18T04:23:02.175282Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:23:02.175282Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VoFMXPC72tIUMdPac8/QpAqBZ4Rrb/QjR9bSNsYMnwyhVNWdkD8JO6tlQn92d1jHGI++1J0qB5abuqPJR3sfDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:23:02.176186Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.6266","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f9eac650e5f1f87bd49012333057db3524daa47f0d92cbb626043b48b9e609e3","sha256:67df1e5e6e8b397e9bf97df37bad16e213b603b564e8ef567b4320ae9cc270d5"],"state_sha256":"9ba23dc87b2dccd1baaa03725ef0bb5640bd1c4a862f72524bac04e3360ed24a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JmgSJuc5A+2OijkC0Bdj5HI5UQEnv9gGYo+EYxPY57ve+My7oNwTK6OrE31l1a/ka/411inJP8zIKqYdQ9jRDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T22:56:49.612649Z","bundle_sha256":"2c628f1560a4fc2ff143bc7a6a9b78cb1e21c25eafc72b47a44fce8da85f4132"}}