{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:ASJKPR55MPG6N6L7WEUQ7C7TOI","short_pith_number":"pith:ASJKPR55","canonical_record":{"source":{"id":"0912.4688","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2009-12-23T16:53:05Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"3eaa13d3640e366893622ce521d2c52f4102ead886f26196a6d2f26c7cfc5300","abstract_canon_sha256":"f1a0d600a704a92eea309b3ba4d42c53718f3bf1b0e482250cad8382b849f54b"},"schema_version":"1.0"},"canonical_sha256":"0492a7c7bd63cde6f97fb1290f8bf37235c8a9bdd8cdf37fe89cd91d68100a04","source":{"kind":"arxiv","id":"0912.4688","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.4688","created_at":"2026-05-18T04:34:01Z"},{"alias_kind":"arxiv_version","alias_value":"0912.4688v2","created_at":"2026-05-18T04:34:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.4688","created_at":"2026-05-18T04:34:01Z"},{"alias_kind":"pith_short_12","alias_value":"ASJKPR55MPG6","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"ASJKPR55MPG6N6L7","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"ASJKPR55","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:ASJKPR55MPG6N6L7WEUQ7C7TOI","target":"record","payload":{"canonical_record":{"source":{"id":"0912.4688","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2009-12-23T16:53:05Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"3eaa13d3640e366893622ce521d2c52f4102ead886f26196a6d2f26c7cfc5300","abstract_canon_sha256":"f1a0d600a704a92eea309b3ba4d42c53718f3bf1b0e482250cad8382b849f54b"},"schema_version":"1.0"},"canonical_sha256":"0492a7c7bd63cde6f97fb1290f8bf37235c8a9bdd8cdf37fe89cd91d68100a04","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:01.451268Z","signature_b64":"hYuBSwkTUcb0h8eqT1pVADiRobRCL2xJtulty93eJqOJfLIoED70mnwUAqfGM2NpESJf4Fn6dxj3gZwYJMduBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0492a7c7bd63cde6f97fb1290f8bf37235c8a9bdd8cdf37fe89cd91d68100a04","last_reissued_at":"2026-05-18T04:34:01.450834Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:01.450834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0912.4688","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":"2026-05-18T04:34:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vQYYf7q4bV2w+02JK9B2R5qFoQQx+AcqJOF6vxmQL9OxbNSOF9iBoSQxZgsBdDl3k3GzNo9o774kiHe3kyFfCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T19:10:43.464963Z"},"content_sha256":"60490f550e230bc8fdc269644fd90d0e1b1796909d516f075803704c3dc2939f","schema_version":"1.0","event_id":"sha256:60490f550e230bc8fdc269644fd90d0e1b1796909d516f075803704c3dc2939f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:ASJKPR55MPG6N6L7WEUQ7C7TOI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotic properties of U-processes under long-range dependence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"C\\'eline L\\'evy-Leduc (LTCI), Eric Moulines (LTCI), H\\'el\\`ene Boistard (GREMAQ), Murad S. Taqqu, Valderio A. Reisen","submitted_at":"2009-12-23T16:53:05Z","abstract_excerpt":"Let $(X_i)_{i\\geq 1}$ be a stationary mean-zero Gaussian process with covariances $\\rho(k)=\\PE(X_{1}X_{k+1})$ satisfying: $\\rho(0)=1$ and $\\rho(k)=k^{-D} L(k)$ where $D$ is in $(0,1)$ and $L$ is slowly varying at infinity. Consider the $U$-process $\\{U_n(r),\\; r\\in I\\}$ defined as $$ U_n(r)=\\frac{1}{n(n-1)}\\sum_{1\\leq i\\neq j\\leq n}\\1_{\\{G(X_i,X_j)\\leq r\\}}\\; , $$ where $I$ is an interval included in $\\rset$ and $G$ is a symmetric function. In this paper, we provide central and non-central limit theorems for $U_n$. They are used to derive the asymptotic behavior of the Hodges-Lehmann estimator"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.4688","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":"2026-05-18T04:34:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P+v2IeiS6nW0RRm5JBT0yvZlBN85bKEIRgU+NkNSjptTRqo+v78oee4Ij4x8h+eRYCZ6NN9hczyjm9aGw5O6Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T19:10:43.465340Z"},"content_sha256":"2dee3523149f1f9dc2d20da5f8fd18bfef2d748b1ad9bdcc22debbcc80991e07","schema_version":"1.0","event_id":"sha256:2dee3523149f1f9dc2d20da5f8fd18bfef2d748b1ad9bdcc22debbcc80991e07"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ASJKPR55MPG6N6L7WEUQ7C7TOI/bundle.json","state_url":"https://pith.science/pith/ASJKPR55MPG6N6L7WEUQ7C7TOI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ASJKPR55MPG6N6L7WEUQ7C7TOI/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":"2026-05-26T19:10:43Z","links":{"resolver":"https://pith.science/pith/ASJKPR55MPG6N6L7WEUQ7C7TOI","bundle":"https://pith.science/pith/ASJKPR55MPG6N6L7WEUQ7C7TOI/bundle.json","state":"https://pith.science/pith/ASJKPR55MPG6N6L7WEUQ7C7TOI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ASJKPR55MPG6N6L7WEUQ7C7TOI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:ASJKPR55MPG6N6L7WEUQ7C7TOI","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":"f1a0d600a704a92eea309b3ba4d42c53718f3bf1b0e482250cad8382b849f54b","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2009-12-23T16:53:05Z","title_canon_sha256":"3eaa13d3640e366893622ce521d2c52f4102ead886f26196a6d2f26c7cfc5300"},"schema_version":"1.0","source":{"id":"0912.4688","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.4688","created_at":"2026-05-18T04:34:01Z"},{"alias_kind":"arxiv_version","alias_value":"0912.4688v2","created_at":"2026-05-18T04:34:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.4688","created_at":"2026-05-18T04:34:01Z"},{"alias_kind":"pith_short_12","alias_value":"ASJKPR55MPG6","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"ASJKPR55MPG6N6L7","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"ASJKPR55","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:2dee3523149f1f9dc2d20da5f8fd18bfef2d748b1ad9bdcc22debbcc80991e07","target":"graph","created_at":"2026-05-18T04:34:01Z","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":"Let $(X_i)_{i\\geq 1}$ be a stationary mean-zero Gaussian process with covariances $\\rho(k)=\\PE(X_{1}X_{k+1})$ satisfying: $\\rho(0)=1$ and $\\rho(k)=k^{-D} L(k)$ where $D$ is in $(0,1)$ and $L$ is slowly varying at infinity. Consider the $U$-process $\\{U_n(r),\\; r\\in I\\}$ defined as $$ U_n(r)=\\frac{1}{n(n-1)}\\sum_{1\\leq i\\neq j\\leq n}\\1_{\\{G(X_i,X_j)\\leq r\\}}\\; , $$ where $I$ is an interval included in $\\rset$ and $G$ is a symmetric function. In this paper, we provide central and non-central limit theorems for $U_n$. They are used to derive the asymptotic behavior of the Hodges-Lehmann estimator","authors_text":"C\\'eline L\\'evy-Leduc (LTCI), Eric Moulines (LTCI), H\\'el\\`ene Boistard (GREMAQ), Murad S. Taqqu, Valderio A. Reisen","cross_cats":["stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2009-12-23T16:53:05Z","title":"Asymptotic properties of U-processes under long-range dependence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.4688","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":"sha256:60490f550e230bc8fdc269644fd90d0e1b1796909d516f075803704c3dc2939f","target":"record","created_at":"2026-05-18T04:34:01Z","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":"f1a0d600a704a92eea309b3ba4d42c53718f3bf1b0e482250cad8382b849f54b","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2009-12-23T16:53:05Z","title_canon_sha256":"3eaa13d3640e366893622ce521d2c52f4102ead886f26196a6d2f26c7cfc5300"},"schema_version":"1.0","source":{"id":"0912.4688","kind":"arxiv","version":2}},"canonical_sha256":"0492a7c7bd63cde6f97fb1290f8bf37235c8a9bdd8cdf37fe89cd91d68100a04","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0492a7c7bd63cde6f97fb1290f8bf37235c8a9bdd8cdf37fe89cd91d68100a04","first_computed_at":"2026-05-18T04:34:01.450834Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:34:01.450834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hYuBSwkTUcb0h8eqT1pVADiRobRCL2xJtulty93eJqOJfLIoED70mnwUAqfGM2NpESJf4Fn6dxj3gZwYJMduBg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:34:01.451268Z","signed_message":"canonical_sha256_bytes"},"source_id":"0912.4688","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:60490f550e230bc8fdc269644fd90d0e1b1796909d516f075803704c3dc2939f","sha256:2dee3523149f1f9dc2d20da5f8fd18bfef2d748b1ad9bdcc22debbcc80991e07"],"state_sha256":"ffd29e21081278687d53499b7147d353119ee3204e6c99004b5c8fdf8c2b7906"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w4ipYffwTkFj4wXnRtglOz7FI1BfzNFeO6CpqgUSaKUga+VC4Qf40T1Jx9+2vulefrEc1S8/1SiufZ0lrB0TCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T19:10:43.467795Z","bundle_sha256":"fd037929f1abdbfbd18e1e0a47cc8eb0b66171a484b4cf2ab5abfb89cb7b3d6b"}}