{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1999:EENCAB6LFY6AFIDO2EX344GV57","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":"7753bfe8dc1fea295ca3e87c9e7f5366aab9ba1252fdd54739482833b8ced95a","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"1999-05-20T17:19:17Z","title_canon_sha256":"1f04031bb76cedd7bf14be1ff95193aa2a837b275dbd36b2b3c701bba82b3c62"},"schema_version":"1.0","source":{"id":"math/9905132","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9905132","created_at":"2026-05-18T02:37:27Z"},{"alias_kind":"arxiv_version","alias_value":"math/9905132v1","created_at":"2026-05-18T02:37:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9905132","created_at":"2026-05-18T02:37:27Z"},{"alias_kind":"pith_short_12","alias_value":"EENCAB6LFY6A","created_at":"2026-05-18T12:25:49Z"},{"alias_kind":"pith_short_16","alias_value":"EENCAB6LFY6AFIDO","created_at":"2026-05-18T12:25:49Z"},{"alias_kind":"pith_short_8","alias_value":"EENCAB6L","created_at":"2026-05-18T12:25:49Z"}],"graph_snapshots":[{"event_id":"sha256:4d7be70192a8b6fb9e3b84a49829dc17ce7838caf134aee9e494855d2b13684a","target":"graph","created_at":"2026-05-18T02:37:27Z","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,X_1,X_2,... be independent identically distributed random variables and let h(x,y)=h(y,x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, $\\limsup_n (n\\log\\log n)^{-1}|\\sum_{1<= i< j<= n}h(X_i,X_j)|<\\infty$ a.s., holds if and only if the following three conditions are satisfied: h is canonical for the law of X (that is Eh(X,y)=0 for almost y) and there exists $C<\\infty$ such that, both, $E\\min(h^2(X_1,X_2),u)<C\\log\\log u$ for all large u and $sup\\{Eh(X_1,X_2)f(X_1)g(X_2):|f(X)|_2<1,\\|g(X)\\|_2<1, \\|f\\|_\\infty<\\infty, \\|g\\|_\\infty<\\inf","authors_text":"Evarist Gin\\'e, Joel Zinn, Rafa{\\l} Lata{\\l}a, Stanis{\\l}aw Kwapie\\'n","cross_cats":[],"headline":"","license":"","primary_cat":"math.PR","submitted_at":"1999-05-20T17:19:17Z","title":"The LIL for canonical U-statistics of order 2"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9905132","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":"sha256:6137eea95e6fde2f09b97bf0531450567a1ed75d9aef04ce64a7cb6901345447","target":"record","created_at":"2026-05-18T02:37:27Z","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":"7753bfe8dc1fea295ca3e87c9e7f5366aab9ba1252fdd54739482833b8ced95a","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"1999-05-20T17:19:17Z","title_canon_sha256":"1f04031bb76cedd7bf14be1ff95193aa2a837b275dbd36b2b3c701bba82b3c62"},"schema_version":"1.0","source":{"id":"math/9905132","kind":"arxiv","version":1}},"canonical_sha256":"211a2007cb2e3c02a06ed12fbe70d5efd31a6f4671b09eb618db159216788794","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"211a2007cb2e3c02a06ed12fbe70d5efd31a6f4671b09eb618db159216788794","first_computed_at":"2026-05-18T02:37:27.788987Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:37:27.788987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"H3NOIAgogl0oatIX64bqRewvVrGPY0Fkt0mnci/FScEexhKG4HFoPffpdBxO2ojKrlrUVSl/Fsi4252A7HXCCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:37:27.791349Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9905132","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6137eea95e6fde2f09b97bf0531450567a1ed75d9aef04ce64a7cb6901345447","sha256:4d7be70192a8b6fb9e3b84a49829dc17ce7838caf134aee9e494855d2b13684a"],"state_sha256":"f0f52f1b4985b6575c3b208744ea06841eb8e4414177d7029a068c9652de9a5a"}