{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:QYD2SOV36ASKEAWT5GFUTVPCVH","short_pith_number":"pith:QYD2SOV3","canonical_record":{"source":{"id":"1507.06406","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-23T08:08:29Z","cross_cats_sorted":[],"title_canon_sha256":"9eebcb99fc1ff36b406b75d47c20b8e6c471527db47684a347ffd7704cdca2ac","abstract_canon_sha256":"b1c7731295fa85feac0c1d6e062737bb03be6b02f988497e98e0f1bb7dd580ec"},"schema_version":"1.0"},"canonical_sha256":"8607a93abbf024a202d3e98b49d5e2a9e637ef247cf19500854647a0050247f6","source":{"kind":"arxiv","id":"1507.06406","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.06406","created_at":"2026-05-18T01:36:25Z"},{"alias_kind":"arxiv_version","alias_value":"1507.06406v1","created_at":"2026-05-18T01:36:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.06406","created_at":"2026-05-18T01:36:25Z"},{"alias_kind":"pith_short_12","alias_value":"QYD2SOV36ASK","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"QYD2SOV36ASKEAWT","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"QYD2SOV3","created_at":"2026-05-18T12:29:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:QYD2SOV36ASKEAWT5GFUTVPCVH","target":"record","payload":{"canonical_record":{"source":{"id":"1507.06406","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-23T08:08:29Z","cross_cats_sorted":[],"title_canon_sha256":"9eebcb99fc1ff36b406b75d47c20b8e6c471527db47684a347ffd7704cdca2ac","abstract_canon_sha256":"b1c7731295fa85feac0c1d6e062737bb03be6b02f988497e98e0f1bb7dd580ec"},"schema_version":"1.0"},"canonical_sha256":"8607a93abbf024a202d3e98b49d5e2a9e637ef247cf19500854647a0050247f6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:25.032987Z","signature_b64":"rV1qCNZ5/wfuiN10TlxX+VrJ32DaeWhJCRKpAMgcaFHWXymhDQAZDo2PIovTq7ZWwuKHELoppTsBZnf9kcybBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8607a93abbf024a202d3e98b49d5e2a9e637ef247cf19500854647a0050247f6","last_reissued_at":"2026-05-18T01:36:25.032562Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:25.032562Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.06406","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":"2026-05-18T01:36:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jWKcR9dm4hf7nL5extssogqgxjFLrN41uRBfMIJxWNTgonSVrs/Ss1gKeoakKFRYNV/aYlJlTTrF4GEq9PwgBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T04:47:00.576864Z"},"content_sha256":"8a0695bc0efe40d5841fdfc3d8a1100947de7061732cec0b233b2e0251df7ae3","schema_version":"1.0","event_id":"sha256:8a0695bc0efe40d5841fdfc3d8a1100947de7061732cec0b233b2e0251df7ae3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:QYD2SOV36ASKEAWT5GFUTVPCVH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Convergence rate in precise asymptotics for Davis law of large numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Lingtao Kong","submitted_at":"2015-07-23T08:08:29Z","abstract_excerpt":"Let $\\{X_n,n\\geq 1\\}$ be a sequence of i.i.d. random variables with partial sums $\\{S_n,n\\geq 1\\}$. Based on the classical Baum-Katz theorem, a paper by Heyde in 1975 initiated the precise asymptotics for the sum $\\sum_{n\\geq 1}\\mbox{P}(|S_n|\\geq\\epsilon n)$ as $\\epsilon$ goes to zero. Later, Klesov determined the convergence rate in Heyde's theorem. The aim of this paper is to extend Klesov's result to the precise asymptotics for Davis law of large numbers, a theorem in Gut and Sp\\u{a}taru [2000a]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06406","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":"2026-05-18T01:36:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GjEJVlU16pZIxlx+g7XWPKxJqiFc5X8DiDF+WuV5V8kIVGtcDQrzloKNeKR99Q+VcK5xaPpOTUzzHRmKwfdbAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T04:47:00.577201Z"},"content_sha256":"1dcf57887437b25b7bbfe748fa9857d29979473c08bb029a78301b9011472f99","schema_version":"1.0","event_id":"sha256:1dcf57887437b25b7bbfe748fa9857d29979473c08bb029a78301b9011472f99"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QYD2SOV36ASKEAWT5GFUTVPCVH/bundle.json","state_url":"https://pith.science/pith/QYD2SOV36ASKEAWT5GFUTVPCVH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QYD2SOV36ASKEAWT5GFUTVPCVH/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":"2026-06-27T04:47:00Z","links":{"resolver":"https://pith.science/pith/QYD2SOV36ASKEAWT5GFUTVPCVH","bundle":"https://pith.science/pith/QYD2SOV36ASKEAWT5GFUTVPCVH/bundle.json","state":"https://pith.science/pith/QYD2SOV36ASKEAWT5GFUTVPCVH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QYD2SOV36ASKEAWT5GFUTVPCVH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:QYD2SOV36ASKEAWT5GFUTVPCVH","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":"b1c7731295fa85feac0c1d6e062737bb03be6b02f988497e98e0f1bb7dd580ec","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-23T08:08:29Z","title_canon_sha256":"9eebcb99fc1ff36b406b75d47c20b8e6c471527db47684a347ffd7704cdca2ac"},"schema_version":"1.0","source":{"id":"1507.06406","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.06406","created_at":"2026-05-18T01:36:25Z"},{"alias_kind":"arxiv_version","alias_value":"1507.06406v1","created_at":"2026-05-18T01:36:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.06406","created_at":"2026-05-18T01:36:25Z"},{"alias_kind":"pith_short_12","alias_value":"QYD2SOV36ASK","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"QYD2SOV36ASKEAWT","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"QYD2SOV3","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:1dcf57887437b25b7bbfe748fa9857d29979473c08bb029a78301b9011472f99","target":"graph","created_at":"2026-05-18T01:36:25Z","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 $\\{X_n,n\\geq 1\\}$ be a sequence of i.i.d. random variables with partial sums $\\{S_n,n\\geq 1\\}$. Based on the classical Baum-Katz theorem, a paper by Heyde in 1975 initiated the precise asymptotics for the sum $\\sum_{n\\geq 1}\\mbox{P}(|S_n|\\geq\\epsilon n)$ as $\\epsilon$ goes to zero. Later, Klesov determined the convergence rate in Heyde's theorem. The aim of this paper is to extend Klesov's result to the precise asymptotics for Davis law of large numbers, a theorem in Gut and Sp\\u{a}taru [2000a].","authors_text":"Lingtao Kong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-23T08:08:29Z","title":"Convergence rate in precise asymptotics for Davis law of large numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06406","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":"sha256:8a0695bc0efe40d5841fdfc3d8a1100947de7061732cec0b233b2e0251df7ae3","target":"record","created_at":"2026-05-18T01:36:25Z","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":"b1c7731295fa85feac0c1d6e062737bb03be6b02f988497e98e0f1bb7dd580ec","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-23T08:08:29Z","title_canon_sha256":"9eebcb99fc1ff36b406b75d47c20b8e6c471527db47684a347ffd7704cdca2ac"},"schema_version":"1.0","source":{"id":"1507.06406","kind":"arxiv","version":1}},"canonical_sha256":"8607a93abbf024a202d3e98b49d5e2a9e637ef247cf19500854647a0050247f6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8607a93abbf024a202d3e98b49d5e2a9e637ef247cf19500854647a0050247f6","first_computed_at":"2026-05-18T01:36:25.032562Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:25.032562Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rV1qCNZ5/wfuiN10TlxX+VrJ32DaeWhJCRKpAMgcaFHWXymhDQAZDo2PIovTq7ZWwuKHELoppTsBZnf9kcybBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:25.032987Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.06406","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8a0695bc0efe40d5841fdfc3d8a1100947de7061732cec0b233b2e0251df7ae3","sha256:1dcf57887437b25b7bbfe748fa9857d29979473c08bb029a78301b9011472f99"],"state_sha256":"9540e29d7358513a4889f1aa23aff5a6ca6a0d3c0c05651671c1a149907521c0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mE9qLRmOGzbJ8u8rH4O/dEm86yy2omX0BVFeI9zQzAAy76wCf5IgibbDcQJ5OTDUCFeT6zCL6eicNRWP1RSmCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T04:47:00.579140Z","bundle_sha256":"3e257b559a150b36fcdc30308537ee98086afa7e2bd603b3fe5d10c0579e77bd"}}