{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:R3G47FBBFKWD2BH2B524F4BIOR","short_pith_number":"pith:R3G47FBB","canonical_record":{"source":{"id":"1212.2133","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-12-10T17:04:25Z","cross_cats_sorted":[],"title_canon_sha256":"a4937728fd50019ee6d548833aa3373dce8060e24459abe2f6b1413dd537220c","abstract_canon_sha256":"12bc9c1c8dbec76adf2c7a0719bdc5bcc495c34867e36edd3ed42c1bb802f18f"},"schema_version":"1.0"},"canonical_sha256":"8ecdcf94212aac3d04fa0f75c2f028745268f497c221a249807010d0a0df5ee9","source":{"kind":"arxiv","id":"1212.2133","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2133","created_at":"2026-05-18T02:25:49Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2133v3","created_at":"2026-05-18T02:25:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2133","created_at":"2026-05-18T02:25:49Z"},{"alias_kind":"pith_short_12","alias_value":"R3G47FBBFKWD","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"R3G47FBBFKWD2BH2","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"R3G47FBB","created_at":"2026-05-18T12:27:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:R3G47FBBFKWD2BH2B524F4BIOR","target":"record","payload":{"canonical_record":{"source":{"id":"1212.2133","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-12-10T17:04:25Z","cross_cats_sorted":[],"title_canon_sha256":"a4937728fd50019ee6d548833aa3373dce8060e24459abe2f6b1413dd537220c","abstract_canon_sha256":"12bc9c1c8dbec76adf2c7a0719bdc5bcc495c34867e36edd3ed42c1bb802f18f"},"schema_version":"1.0"},"canonical_sha256":"8ecdcf94212aac3d04fa0f75c2f028745268f497c221a249807010d0a0df5ee9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:49.366751Z","signature_b64":"I+rYaA1ChMBNK4QbWM0WyG6Cmer9kAvtl+IcWWx5GtQ3dIP/yonpVFJZZ3OcAhduYUMkn33JUuguZZ2BIBToCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ecdcf94212aac3d04fa0f75c2f028745268f497c221a249807010d0a0df5ee9","last_reissued_at":"2026-05-18T02:25:49.366359Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:49.366359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.2133","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:25:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N5qz422zbEex3joNcQd0lO/ltzIaVP6JjfYZzAjJHRXdHMPxcZmXi/NwH9vqh+fsmHBH77P9/f12zFts6ReTAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T21:25:32.218503Z"},"content_sha256":"40c11f218f648d926443bacdc8b05173d38405b40761b0b4c3dd4e159a852676","schema_version":"1.0","event_id":"sha256:40c11f218f648d926443bacdc8b05173d38405b40761b0b4c3dd4e159a852676"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:R3G47FBBFKWD2BH2B524F4BIOR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stable Limit Theorem for U-Statistic Processes Indexed by a Random Walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Brice Franke, Francoise Pene, Martin Wendler","submitted_at":"2012-12-10T17:04:25Z","abstract_excerpt":"Let (S_n)_{n\\in\\N} be a Z-valued random walk with increments from the domain of attraction of some \\alpha-stable law and let (\\xi(i))_{i\\in\\Z} be a sequence of iid random variables. We want to investigate U-statistics indexed by the random walk S_n, that is U_n:=\\sum_{1\\leq i<j\\leq n}h(\\xi(S_i),\\xi(S_j)) for some symmetric bivariate function h. We will prove the weak convergence without assumption of finite variance. Additionally, under the assumption of finite moments of order greater than two, we will establish a law of the iterated logarithm for the U-statistic U_n."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2133","kind":"arxiv","version":3},"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":"2026-05-18T02:25:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y7sV7/lB6b3WW7vHCcU34sXV+G+6K1rKT00S3vpXH0C1ur7Gw9wUcFWX9qD22RkMqo0E64GZH3ML6GLxnxjUAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T21:25:32.218841Z"},"content_sha256":"2655249c05daf75a17f327c6bb3afca7891d5647a22a77f511a28ec898ecc2d7","schema_version":"1.0","event_id":"sha256:2655249c05daf75a17f327c6bb3afca7891d5647a22a77f511a28ec898ecc2d7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/R3G47FBBFKWD2BH2B524F4BIOR/bundle.json","state_url":"https://pith.science/pith/R3G47FBBFKWD2BH2B524F4BIOR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/R3G47FBBFKWD2BH2B524F4BIOR/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":"2026-06-26T21:25:32Z","links":{"resolver":"https://pith.science/pith/R3G47FBBFKWD2BH2B524F4BIOR","bundle":"https://pith.science/pith/R3G47FBBFKWD2BH2B524F4BIOR/bundle.json","state":"https://pith.science/pith/R3G47FBBFKWD2BH2B524F4BIOR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/R3G47FBBFKWD2BH2B524F4BIOR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:R3G47FBBFKWD2BH2B524F4BIOR","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":"12bc9c1c8dbec76adf2c7a0719bdc5bcc495c34867e36edd3ed42c1bb802f18f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-12-10T17:04:25Z","title_canon_sha256":"a4937728fd50019ee6d548833aa3373dce8060e24459abe2f6b1413dd537220c"},"schema_version":"1.0","source":{"id":"1212.2133","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2133","created_at":"2026-05-18T02:25:49Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2133v3","created_at":"2026-05-18T02:25:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2133","created_at":"2026-05-18T02:25:49Z"},{"alias_kind":"pith_short_12","alias_value":"R3G47FBBFKWD","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"R3G47FBBFKWD2BH2","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"R3G47FBB","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:2655249c05daf75a17f327c6bb3afca7891d5647a22a77f511a28ec898ecc2d7","target":"graph","created_at":"2026-05-18T02:25:49Z","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 (S_n)_{n\\in\\N} be a Z-valued random walk with increments from the domain of attraction of some \\alpha-stable law and let (\\xi(i))_{i\\in\\Z} be a sequence of iid random variables. We want to investigate U-statistics indexed by the random walk S_n, that is U_n:=\\sum_{1\\leq i<j\\leq n}h(\\xi(S_i),\\xi(S_j)) for some symmetric bivariate function h. We will prove the weak convergence without assumption of finite variance. Additionally, under the assumption of finite moments of order greater than two, we will establish a law of the iterated logarithm for the U-statistic U_n.","authors_text":"Brice Franke, Francoise Pene, Martin Wendler","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-12-10T17:04:25Z","title":"Stable Limit Theorem for U-Statistic Processes Indexed by a Random Walk"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2133","kind":"arxiv","version":3},"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":"sha256:40c11f218f648d926443bacdc8b05173d38405b40761b0b4c3dd4e159a852676","target":"record","created_at":"2026-05-18T02:25:49Z","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":"12bc9c1c8dbec76adf2c7a0719bdc5bcc495c34867e36edd3ed42c1bb802f18f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-12-10T17:04:25Z","title_canon_sha256":"a4937728fd50019ee6d548833aa3373dce8060e24459abe2f6b1413dd537220c"},"schema_version":"1.0","source":{"id":"1212.2133","kind":"arxiv","version":3}},"canonical_sha256":"8ecdcf94212aac3d04fa0f75c2f028745268f497c221a249807010d0a0df5ee9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ecdcf94212aac3d04fa0f75c2f028745268f497c221a249807010d0a0df5ee9","first_computed_at":"2026-05-18T02:25:49.366359Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:25:49.366359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"I+rYaA1ChMBNK4QbWM0WyG6Cmer9kAvtl+IcWWx5GtQ3dIP/yonpVFJZZ3OcAhduYUMkn33JUuguZZ2BIBToCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:25:49.366751Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.2133","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40c11f218f648d926443bacdc8b05173d38405b40761b0b4c3dd4e159a852676","sha256:2655249c05daf75a17f327c6bb3afca7891d5647a22a77f511a28ec898ecc2d7"],"state_sha256":"0f17a3c2f0643fc80df0fb49ce06b3ac0153f47d29dee795368522b8d2caa3ee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B3XWHdyzrZE6zSbWwHCBD4cRhXX/1yUh9gR9+aEjleHPkPQ9xwr6GGUSo5olfBYWfLl6AAbQPsM7Fmz5aSaOAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T21:25:32.220681Z","bundle_sha256":"a6e546d5ca8bead1e0a8042802a91b3d470d9920ff82d11213708fc0d5e23f7d"}}