{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:3WMUJCHJLHWXZ2DWP3XYOW4YNK","short_pith_number":"pith:3WMUJCHJ","schema_version":"1.0","canonical_sha256":"dd994488e959ed7ce8767eef875b986aa9647885b994e9abdfc5408da5c45230","source":{"kind":"arxiv","id":"1007.1738","version":2},"attestation_state":"computed","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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1007.1738","created_at":"2026-05-18T03:33:26.899296+00:00"},{"alias_kind":"arxiv_version","alias_value":"1007.1738v2","created_at":"2026-05-18T03:33:26.899296+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.1738","created_at":"2026-05-18T03:33:26.899296+00:00"},{"alias_kind":"pith_short_12","alias_value":"3WMUJCHJLHWX","created_at":"2026-05-18T12:26:03.138858+00:00"},{"alias_kind":"pith_short_16","alias_value":"3WMUJCHJLHWXZ2DW","created_at":"2026-05-18T12:26:03.138858+00:00"},{"alias_kind":"pith_short_8","alias_value":"3WMUJCHJ","created_at":"2026-05-18T12:26:03.138858+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK","json":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK.json","graph_json":"https://pith.science/api/pith-number/3WMUJCHJLHWXZ2DWP3XYOW4YNK/graph.json","events_json":"https://pith.science/api/pith-number/3WMUJCHJLHWXZ2DWP3XYOW4YNK/events.json","paper":"https://pith.science/paper/3WMUJCHJ"},"agent_actions":{"view_html":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK","download_json":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK.json","view_paper":"https://pith.science/paper/3WMUJCHJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1007.1738&json=true","fetch_graph":"https://pith.science/api/pith-number/3WMUJCHJLHWXZ2DWP3XYOW4YNK/graph.json","fetch_events":"https://pith.science/api/pith-number/3WMUJCHJLHWXZ2DWP3XYOW4YNK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK/action/storage_attestation","attest_author":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK/action/author_attestation","sign_citation":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK/action/citation_signature","submit_replication":"https://pith.science/pith/3WMUJCHJLHWXZ2DWP3XYOW4YNK/action/replication_record"}},"created_at":"2026-05-18T03:33:26.899296+00:00","updated_at":"2026-05-18T03:33:26.899296+00:00"}