{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:2SOZMR4Y24TJYDROQ76MZYS752","short_pith_number":"pith:2SOZMR4Y","canonical_record":{"source":{"id":"1303.3029","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-12T21:01:38Z","cross_cats_sorted":[],"title_canon_sha256":"199d9cbf07699c603f3e8341a340031d599191aac600cdf128407614fceb5a24","abstract_canon_sha256":"a135e323a337ba7f363120dd94fb73730285bdae52d2fb441ac2f1f73183a49c"},"schema_version":"1.0"},"canonical_sha256":"d49d964798d7269c0e2e87fccce25fee95ef5da1a00832440777034a73c1b311","source":{"kind":"arxiv","id":"1303.3029","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.3029","created_at":"2026-05-18T03:31:06Z"},{"alias_kind":"arxiv_version","alias_value":"1303.3029v1","created_at":"2026-05-18T03:31:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3029","created_at":"2026-05-18T03:31:06Z"},{"alias_kind":"pith_short_12","alias_value":"2SOZMR4Y24TJ","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"2SOZMR4Y24TJYDRO","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"2SOZMR4Y","created_at":"2026-05-18T12:27:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:2SOZMR4Y24TJYDROQ76MZYS752","target":"record","payload":{"canonical_record":{"source":{"id":"1303.3029","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-12T21:01:38Z","cross_cats_sorted":[],"title_canon_sha256":"199d9cbf07699c603f3e8341a340031d599191aac600cdf128407614fceb5a24","abstract_canon_sha256":"a135e323a337ba7f363120dd94fb73730285bdae52d2fb441ac2f1f73183a49c"},"schema_version":"1.0"},"canonical_sha256":"d49d964798d7269c0e2e87fccce25fee95ef5da1a00832440777034a73c1b311","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:31:06.003936Z","signature_b64":"EUBK+lUxdOxszG/qMx8+I15OwgJ/7/gFIXtbQS4N1isZ60jf4mwSs7DzIN+tklySHFcjhbea8d1ef17e8xXVCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d49d964798d7269c0e2e87fccce25fee95ef5da1a00832440777034a73c1b311","last_reissued_at":"2026-05-18T03:31:06.003211Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:31:06.003211Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1303.3029","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":"2026-05-18T03:31:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TjEAylpDqW72UIVg4ysjCYn7Zv+euHZz1mtTBfslsWRvAWhN1t8skloia7Ma+7PZ5pAyykEJV1tuZpIjP5nDCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:07:59.868078Z"},"content_sha256":"463fc35be8070221771a959a11198f12930cf7f8c537b358006a55869f5c4ba1","schema_version":"1.0","event_id":"sha256:463fc35be8070221771a959a11198f12930cf7f8c537b358006a55869f5c4ba1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:2SOZMR4Y24TJYDROQ76MZYS752","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Theory of approximation and continuity of random processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"E.Ostrovsky, L.Sirota","submitted_at":"2013-03-12T21:01:38Z","abstract_excerpt":"We introduce and investigate a new notion of the theory of approximation-the so-called degenerate approximation, i.e. approximation of the function of two (and more) variables (kernel) by means of degenerate function (kernel).\n  We apply obtained results to the investigation of the local structure of random processes, for example, we find the necessary and sufficient condition for continuity of Gaussian and non-Gaussian processes, some conditions for weak compactness and convergence of a family of random processes, in particular, for Central Limit Theorem in the space of continuous functions.\n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3029","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":"2026-05-18T03:31:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PKI/HF3H6OwgkVp303FU/nO8duvMxNvDPt7CE30R5kMz8Bk7FWyJsoUCMBi1/Dlh9MvzKFlkqcpothNGes1sDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:07:59.868481Z"},"content_sha256":"1c239784b7178df82d89105ccd966ea1ec20260b396529d0437b7c7a84f4c3d2","schema_version":"1.0","event_id":"sha256:1c239784b7178df82d89105ccd966ea1ec20260b396529d0437b7c7a84f4c3d2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2SOZMR4Y24TJYDROQ76MZYS752/bundle.json","state_url":"https://pith.science/pith/2SOZMR4Y24TJYDROQ76MZYS752/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2SOZMR4Y24TJYDROQ76MZYS752/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":"2026-06-23T03:07:59Z","links":{"resolver":"https://pith.science/pith/2SOZMR4Y24TJYDROQ76MZYS752","bundle":"https://pith.science/pith/2SOZMR4Y24TJYDROQ76MZYS752/bundle.json","state":"https://pith.science/pith/2SOZMR4Y24TJYDROQ76MZYS752/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2SOZMR4Y24TJYDROQ76MZYS752/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:2SOZMR4Y24TJYDROQ76MZYS752","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":"a135e323a337ba7f363120dd94fb73730285bdae52d2fb441ac2f1f73183a49c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-12T21:01:38Z","title_canon_sha256":"199d9cbf07699c603f3e8341a340031d599191aac600cdf128407614fceb5a24"},"schema_version":"1.0","source":{"id":"1303.3029","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.3029","created_at":"2026-05-18T03:31:06Z"},{"alias_kind":"arxiv_version","alias_value":"1303.3029v1","created_at":"2026-05-18T03:31:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3029","created_at":"2026-05-18T03:31:06Z"},{"alias_kind":"pith_short_12","alias_value":"2SOZMR4Y24TJ","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"2SOZMR4Y24TJYDRO","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"2SOZMR4Y","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:1c239784b7178df82d89105ccd966ea1ec20260b396529d0437b7c7a84f4c3d2","target":"graph","created_at":"2026-05-18T03:31:06Z","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":"We introduce and investigate a new notion of the theory of approximation-the so-called degenerate approximation, i.e. approximation of the function of two (and more) variables (kernel) by means of degenerate function (kernel).\n  We apply obtained results to the investigation of the local structure of random processes, for example, we find the necessary and sufficient condition for continuity of Gaussian and non-Gaussian processes, some conditions for weak compactness and convergence of a family of random processes, in particular, for Central Limit Theorem in the space of continuous functions.\n","authors_text":"E.Ostrovsky, L.Sirota","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-12T21:01:38Z","title":"Theory of approximation and continuity of random processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3029","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":"sha256:463fc35be8070221771a959a11198f12930cf7f8c537b358006a55869f5c4ba1","target":"record","created_at":"2026-05-18T03:31:06Z","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":"a135e323a337ba7f363120dd94fb73730285bdae52d2fb441ac2f1f73183a49c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-12T21:01:38Z","title_canon_sha256":"199d9cbf07699c603f3e8341a340031d599191aac600cdf128407614fceb5a24"},"schema_version":"1.0","source":{"id":"1303.3029","kind":"arxiv","version":1}},"canonical_sha256":"d49d964798d7269c0e2e87fccce25fee95ef5da1a00832440777034a73c1b311","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d49d964798d7269c0e2e87fccce25fee95ef5da1a00832440777034a73c1b311","first_computed_at":"2026-05-18T03:31:06.003211Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:31:06.003211Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EUBK+lUxdOxszG/qMx8+I15OwgJ/7/gFIXtbQS4N1isZ60jf4mwSs7DzIN+tklySHFcjhbea8d1ef17e8xXVCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:31:06.003936Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.3029","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:463fc35be8070221771a959a11198f12930cf7f8c537b358006a55869f5c4ba1","sha256:1c239784b7178df82d89105ccd966ea1ec20260b396529d0437b7c7a84f4c3d2"],"state_sha256":"56a68d1bd788d0af545f49ecc9da62a23b0217d761a16e441b51ebc0202773ef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PvnbmoGkYCms/xxGrHUlF3hj+X7lXJfu2F4mzNu0M7TMNmBhR2qqUjVbVdS1A+bm69TiykZrr33NW7NqGA82Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T03:07:59.870475Z","bundle_sha256":"d36d2f17cb2fc7b4ba64507db8c2298d917c0f983f647196ac362c0fa0e624a4"}}