{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:3WMUJCHJLHWXZ2DWP3XYOW4YNK","short_pith_number":"pith:3WMUJCHJ","canonical_record":{"source":{"id":"1007.1738","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-07-10T19:39:40Z","cross_cats_sorted":[],"title_canon_sha256":"c00b74d851f3a6c0dae95380e25fbbc3847894ef48141edbef67da4bdd8b6676","abstract_canon_sha256":"c56871d8239fba84fe632be2240b956904a0ad49fb01bf815a71a006ac266b56"},"schema_version":"1.0"},"canonical_sha256":"dd994488e959ed7ce8767eef875b986aa9647885b994e9abdfc5408da5c45230","source":{"kind":"arxiv","id":"1007.1738","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.1738","created_at":"2026-05-18T03:33:26Z"},{"alias_kind":"arxiv_version","alias_value":"1007.1738v2","created_at":"2026-05-18T03:33:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.1738","created_at":"2026-05-18T03:33:26Z"},{"alias_kind":"pith_short_12","alias_value":"3WMUJCHJLHWX","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"3WMUJCHJLHWXZ2DW","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"3WMUJCHJ","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:3WMUJCHJLHWXZ2DWP3XYOW4YNK","target":"record","payload":{"canonical_record":{"source":{"id":"1007.1738","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-07-10T19:39:40Z","cross_cats_sorted":[],"title_canon_sha256":"c00b74d851f3a6c0dae95380e25fbbc3847894ef48141edbef67da4bdd8b6676","abstract_canon_sha256":"c56871d8239fba84fe632be2240b956904a0ad49fb01bf815a71a006ac266b56"},"schema_version":"1.0"},"canonical_sha256":"dd994488e959ed7ce8767eef875b986aa9647885b994e9abdfc5408da5c45230","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:26.899883Z","signature_b64":"Qidj7QO/ADsOF0iFkhrEuu4QBloZj/kL4pJIVMglO7MbBYOgT5IVf7Io2AMHvo7zsmCFwJVngrubtYXKZDXUBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd994488e959ed7ce8767eef875b986aa9647885b994e9abdfc5408da5c45230","last_reissued_at":"2026-05-18T03:33:26.899181Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:26.899181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.1738","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":"2026-05-18T03:33:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mdRtc2HtJRDzWOhCFKCQHDsyxS2BrpH0iWUoNoO2hDrDTI3DGTBjISLtdn/aYqbYBLGVhBXWzhfZg8cKAc8OCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T02:06:17.409039Z"},"content_sha256":"806ce4d23fc7400cae20189aa261a8b990ebf3386de5f8e80190ba8bb8a960a1","schema_version":"1.0","event_id":"sha256:806ce4d23fc7400cae20189aa261a8b990ebf3386de5f8e80190ba8bb8a960a1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:3WMUJCHJLHWXZ2DWP3XYOW4YNK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Moments, moderate and large deviations for a branching process in a random environment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chunmao Huang, Quansheng Liu","submitted_at":"2010-07-10T19:39:40Z","abstract_excerpt":"Let $(Z_{n})$ be a supercritical branching process in a random environment $\\xi $, and $W$ be the limit of the normalized population size $Z_{n}/\\mathbb{E}[Z_{n}|\\xi ]$. We show large and moderate deviation principles for the sequence $\\log Z_{n}$ (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of $W$, and show an equivalence for all the moments of $Z_{n}$. Central limit theorems on $W-W_n$ and $\\log Z_n$ are also established."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1738","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":"2026-05-18T03:33:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1cyOY+VUlEMDjnpFXU66vK9n836IHdWFxx9g+MiQ4NOIEyU0izlpJ5riIEJWsACj3FdRdWuE0fW0//Gj5m5mCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T02:06:17.409401Z"},"content_sha256":"62d7c0a086c2d62cd9920ae4e32124afe2567abb57a9afa4e6062f5df2b38cc0","schema_version":"1.0","event_id":"sha256:62d7c0a086c2d62cd9920ae4e32124afe2567abb57a9afa4e6062f5df2b38cc0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK/bundle.json","state_url":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK/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":"2026-06-23T02:06:17Z","links":{"resolver":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK","bundle":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK/bundle.json","state":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:3WMUJCHJLHWXZ2DWP3XYOW4YNK","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":"c56871d8239fba84fe632be2240b956904a0ad49fb01bf815a71a006ac266b56","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-07-10T19:39:40Z","title_canon_sha256":"c00b74d851f3a6c0dae95380e25fbbc3847894ef48141edbef67da4bdd8b6676"},"schema_version":"1.0","source":{"id":"1007.1738","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.1738","created_at":"2026-05-18T03:33:26Z"},{"alias_kind":"arxiv_version","alias_value":"1007.1738v2","created_at":"2026-05-18T03:33:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.1738","created_at":"2026-05-18T03:33:26Z"},{"alias_kind":"pith_short_12","alias_value":"3WMUJCHJLHWX","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"3WMUJCHJLHWXZ2DW","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"3WMUJCHJ","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:62d7c0a086c2d62cd9920ae4e32124afe2567abb57a9afa4e6062f5df2b38cc0","target":"graph","created_at":"2026-05-18T03:33:26Z","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 $(Z_{n})$ be a supercritical branching process in a random environment $\\xi $, and $W$ be the limit of the normalized population size $Z_{n}/\\mathbb{E}[Z_{n}|\\xi ]$. We show large and moderate deviation principles for the sequence $\\log Z_{n}$ (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of $W$, and show an equivalence for all the moments of $Z_{n}$. Central limit theorems on $W-W_n$ and $\\log Z_n$ are also established.","authors_text":"Chunmao Huang, Quansheng Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-07-10T19:39:40Z","title":"Moments, moderate and large deviations for a branching process in a random environment"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1738","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":"sha256:806ce4d23fc7400cae20189aa261a8b990ebf3386de5f8e80190ba8bb8a960a1","target":"record","created_at":"2026-05-18T03:33:26Z","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":"c56871d8239fba84fe632be2240b956904a0ad49fb01bf815a71a006ac266b56","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-07-10T19:39:40Z","title_canon_sha256":"c00b74d851f3a6c0dae95380e25fbbc3847894ef48141edbef67da4bdd8b6676"},"schema_version":"1.0","source":{"id":"1007.1738","kind":"arxiv","version":2}},"canonical_sha256":"dd994488e959ed7ce8767eef875b986aa9647885b994e9abdfc5408da5c45230","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dd994488e959ed7ce8767eef875b986aa9647885b994e9abdfc5408da5c45230","first_computed_at":"2026-05-18T03:33:26.899181Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:33:26.899181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Qidj7QO/ADsOF0iFkhrEuu4QBloZj/kL4pJIVMglO7MbBYOgT5IVf7Io2AMHvo7zsmCFwJVngrubtYXKZDXUBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:33:26.899883Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.1738","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:806ce4d23fc7400cae20189aa261a8b990ebf3386de5f8e80190ba8bb8a960a1","sha256:62d7c0a086c2d62cd9920ae4e32124afe2567abb57a9afa4e6062f5df2b38cc0"],"state_sha256":"ed47c12963efac8a2a8eeb886d234a4f37431476a6531526d9829469115c0e46"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mgMgFAauHeT6OFtTTuO3pc/DLsB1dPa/gmuUvP9idzGTq1X9j52ET7W4IhtRjZwVz1PXpRJiUolNL72dQg7YDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T02:06:17.411335Z","bundle_sha256":"755db52887bf9ad3f6548de1b28e767eb484483fb735721cd3221a92393f1780"}}