{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:DFCMY2IGH5FNRVIHAA7CCZB7TL","short_pith_number":"pith:DFCMY2IG","canonical_record":{"source":{"id":"1703.07903","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-23T01:09:42Z","cross_cats_sorted":[],"title_canon_sha256":"f3599ebb6ec1f35e930fbde94454bdf8d3dab2815801d1c0958363d0eeecb49a","abstract_canon_sha256":"d6f236679757dea521d6b48161dfacdc9be6b5df990b0c74f0211affd527fbc8"},"schema_version":"1.0"},"canonical_sha256":"1944cc69063f4ad8d507003e21643f9ac1a98d94bd60ce838dcd028af6793ee1","source":{"kind":"arxiv","id":"1703.07903","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.07903","created_at":"2026-05-18T00:36:38Z"},{"alias_kind":"arxiv_version","alias_value":"1703.07903v2","created_at":"2026-05-18T00:36:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.07903","created_at":"2026-05-18T00:36:38Z"},{"alias_kind":"pith_short_12","alias_value":"DFCMY2IGH5FN","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_16","alias_value":"DFCMY2IGH5FNRVIH","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_8","alias_value":"DFCMY2IG","created_at":"2026-05-18T12:31:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:DFCMY2IGH5FNRVIHAA7CCZB7TL","target":"record","payload":{"canonical_record":{"source":{"id":"1703.07903","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-23T01:09:42Z","cross_cats_sorted":[],"title_canon_sha256":"f3599ebb6ec1f35e930fbde94454bdf8d3dab2815801d1c0958363d0eeecb49a","abstract_canon_sha256":"d6f236679757dea521d6b48161dfacdc9be6b5df990b0c74f0211affd527fbc8"},"schema_version":"1.0"},"canonical_sha256":"1944cc69063f4ad8d507003e21643f9ac1a98d94bd60ce838dcd028af6793ee1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:38.826272Z","signature_b64":"7AiOME7JMQ5gWBdY3dz9mc4o5RKVca3AStv623/7hYuErSpktUj+xGMwcCBmV7DD+BdSJNWzM/Y+ji7RD6NKBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1944cc69063f4ad8d507003e21643f9ac1a98d94bd60ce838dcd028af6793ee1","last_reissued_at":"2026-05-18T00:36:38.825764Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:38.825764Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1703.07903","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":"2026-05-18T00:36:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bNX6IkaxNmepMNtkEXFJKqx0x4iuak4q6yLHpXjAnX/jQAzyt1chrPrTC8MVV9D43nVfNkwaoLQFu5iERl3wAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T22:48:04.529733Z"},"content_sha256":"69a9b421d430df3a5dca4c10688185e204846ab6ec2871a41b4128fdc26f64ef","schema_version":"1.0","event_id":"sha256:69a9b421d430df3a5dca4c10688185e204846ab6ec2871a41b4128fdc26f64ef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:DFCMY2IGH5FNRVIHAA7CCZB7TL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Central limit theorem for Fourier transform and periodogram of random fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Magda Peligrad, Na Zhang","submitted_at":"2017-03-23T01:09:42Z","abstract_excerpt":"In this paper we show that the limiting distribution of the real and the imaginary part of the double Fourier transform of a stationary random field is almost surely an independent vector with Gaussian marginal distributions, whose variance is, up to a constant, the field's spectral density. The dependence structure of the random field is general and we do not impose any restrictions on the speed of convergence to zero of the covariances, or smoothness of the spectral density. The only condition required is that the variables are adapted to a commuting filtration and are regular in some sense."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.07903","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":"2026-05-18T00:36:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4S3wRcXTZ1OcK5gqjDF/nmFmERWniDAKp+H14tkgJslx3Ig+B8OHypgbYDP7KP/n5e/5XqzOE3+x68bNJjDhDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T22:48:04.530390Z"},"content_sha256":"b249a95f3564f5552e2abf4193d3e53698f4e64e3bc5d50bdc0574fc9d1662ad","schema_version":"1.0","event_id":"sha256:b249a95f3564f5552e2abf4193d3e53698f4e64e3bc5d50bdc0574fc9d1662ad"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DFCMY2IGH5FNRVIHAA7CCZB7TL/bundle.json","state_url":"https://pith.science/pith/DFCMY2IGH5FNRVIHAA7CCZB7TL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DFCMY2IGH5FNRVIHAA7CCZB7TL/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":"2026-05-25T22:48:04Z","links":{"resolver":"https://pith.science/pith/DFCMY2IGH5FNRVIHAA7CCZB7TL","bundle":"https://pith.science/pith/DFCMY2IGH5FNRVIHAA7CCZB7TL/bundle.json","state":"https://pith.science/pith/DFCMY2IGH5FNRVIHAA7CCZB7TL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DFCMY2IGH5FNRVIHAA7CCZB7TL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:DFCMY2IGH5FNRVIHAA7CCZB7TL","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":"d6f236679757dea521d6b48161dfacdc9be6b5df990b0c74f0211affd527fbc8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-23T01:09:42Z","title_canon_sha256":"f3599ebb6ec1f35e930fbde94454bdf8d3dab2815801d1c0958363d0eeecb49a"},"schema_version":"1.0","source":{"id":"1703.07903","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.07903","created_at":"2026-05-18T00:36:38Z"},{"alias_kind":"arxiv_version","alias_value":"1703.07903v2","created_at":"2026-05-18T00:36:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.07903","created_at":"2026-05-18T00:36:38Z"},{"alias_kind":"pith_short_12","alias_value":"DFCMY2IGH5FN","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_16","alias_value":"DFCMY2IGH5FNRVIH","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_8","alias_value":"DFCMY2IG","created_at":"2026-05-18T12:31:10Z"}],"graph_snapshots":[{"event_id":"sha256:b249a95f3564f5552e2abf4193d3e53698f4e64e3bc5d50bdc0574fc9d1662ad","target":"graph","created_at":"2026-05-18T00:36:38Z","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 show that the limiting distribution of the real and the imaginary part of the double Fourier transform of a stationary random field is almost surely an independent vector with Gaussian marginal distributions, whose variance is, up to a constant, the field's spectral density. The dependence structure of the random field is general and we do not impose any restrictions on the speed of convergence to zero of the covariances, or smoothness of the spectral density. The only condition required is that the variables are adapted to a commuting filtration and are regular in some sense.","authors_text":"Magda Peligrad, Na Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-23T01:09:42Z","title":"Central limit theorem for Fourier transform and periodogram of random fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.07903","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":"sha256:69a9b421d430df3a5dca4c10688185e204846ab6ec2871a41b4128fdc26f64ef","target":"record","created_at":"2026-05-18T00:36:38Z","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":"d6f236679757dea521d6b48161dfacdc9be6b5df990b0c74f0211affd527fbc8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-23T01:09:42Z","title_canon_sha256":"f3599ebb6ec1f35e930fbde94454bdf8d3dab2815801d1c0958363d0eeecb49a"},"schema_version":"1.0","source":{"id":"1703.07903","kind":"arxiv","version":2}},"canonical_sha256":"1944cc69063f4ad8d507003e21643f9ac1a98d94bd60ce838dcd028af6793ee1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1944cc69063f4ad8d507003e21643f9ac1a98d94bd60ce838dcd028af6793ee1","first_computed_at":"2026-05-18T00:36:38.825764Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:36:38.825764Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7AiOME7JMQ5gWBdY3dz9mc4o5RKVca3AStv623/7hYuErSpktUj+xGMwcCBmV7DD+BdSJNWzM/Y+ji7RD6NKBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:36:38.826272Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.07903","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:69a9b421d430df3a5dca4c10688185e204846ab6ec2871a41b4128fdc26f64ef","sha256:b249a95f3564f5552e2abf4193d3e53698f4e64e3bc5d50bdc0574fc9d1662ad"],"state_sha256":"bf580df676d9b4a539ae38c3942c0e7a9b585ecbb921f5416a6cdf1977c17bfd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aav2JPZF1pbJRfwpqrxJvAlEzGyWY1ygxgSaADcJEU3lRgptMpYEJXf+ZZPoXsfzCtDrpiPbntt+L2O6Hcg4DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T22:48:04.533906Z","bundle_sha256":"60e504f61191686fae1e9c201b12f6caea79f2a230cf6baab64341e384f2fa35"}}