{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:2XP2SLC5I22HV2QKGRKRXOM6W4","short_pith_number":"pith:2XP2SLC5","schema_version":"1.0","canonical_sha256":"d5dfa92c5d46b47aea0a34551bb99eb72bb03764d8a63a8ecae20b560efc55b9","source":{"kind":"arxiv","id":"1811.07317","version":1},"attestation_state":"computed","paper":{"title":"Asymptotic behaviour of heavy-tailed branching processes in random environments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Wenming Hong, Xiaoyue Zhang","submitted_at":"2018-11-18T12:05:26Z","abstract_excerpt":"Consider a heavy-tailed branching process (denoted by $Z_{n}$) in random environments, under the condition which infers that $\\mathbb{E}\\log m(\\xi_{0})=\\infty$. We show that (1) there exists no proper $c_{n}$ such that $\\{Z_{n}/c_{n}\\}$ has a proper, non-degenerate limit, (2) normalized by a sequence of functions, a proper limit can be obtained, i.e., $y_{n}\\left(\\bar{\\xi},Z_{n}(\\bar{\\xi})\\right)$ converges almost surely to a random variable $Y(\\bar{\\xi})$, where $Y\\in(0,1)~\\eta$-a.s., (3) finally, we give a necessary and sufficient conditions for the almost sure convergence of $\\left\\{\\frac{U"},"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":"1811.07317","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-11-18T12:05:26Z","cross_cats_sorted":[],"title_canon_sha256":"316c430e6a2fba56b79aec752df9b8efa7417ff159abe3d37ef4616669229f4e","abstract_canon_sha256":"242ba6f9fc60c3912a1e19ea1697f79656263da5d86a0f83080bee0b6a362a9c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:28.296733Z","signature_b64":"R4YwgJLYJfcpqHhbBCkAwy1IBMbFL4ltG+ViQRJyppezpaDLiiQD6TSiIsNgdrG+Ay3Dh41CyNKUUXarRmSaDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d5dfa92c5d46b47aea0a34551bb99eb72bb03764d8a63a8ecae20b560efc55b9","last_reissued_at":"2026-05-18T00:00:28.296096Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:28.296096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic behaviour of heavy-tailed branching processes in random environments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Wenming Hong, Xiaoyue Zhang","submitted_at":"2018-11-18T12:05:26Z","abstract_excerpt":"Consider a heavy-tailed branching process (denoted by $Z_{n}$) in random environments, under the condition which infers that $\\mathbb{E}\\log m(\\xi_{0})=\\infty$. We show that (1) there exists no proper $c_{n}$ such that $\\{Z_{n}/c_{n}\\}$ has a proper, non-degenerate limit, (2) normalized by a sequence of functions, a proper limit can be obtained, i.e., $y_{n}\\left(\\bar{\\xi},Z_{n}(\\bar{\\xi})\\right)$ converges almost surely to a random variable $Y(\\bar{\\xi})$, where $Y\\in(0,1)~\\eta$-a.s., (3) finally, we give a necessary and sufficient conditions for the almost sure convergence of $\\left\\{\\frac{U"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.07317","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1811.07317","created_at":"2026-05-18T00:00:28.296206+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.07317v1","created_at":"2026-05-18T00:00:28.296206+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.07317","created_at":"2026-05-18T00:00:28.296206+00:00"},{"alias_kind":"pith_short_12","alias_value":"2XP2SLC5I22H","created_at":"2026-05-18T12:32:02.567920+00:00"},{"alias_kind":"pith_short_16","alias_value":"2XP2SLC5I22HV2QK","created_at":"2026-05-18T12:32:02.567920+00:00"},{"alias_kind":"pith_short_8","alias_value":"2XP2SLC5","created_at":"2026-05-18T12:32:02.567920+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/2XP2SLC5I22HV2QKGRKRXOM6W4","json":"https://pith.science/pith/2XP2SLC5I22HV2QKGRKRXOM6W4.json","graph_json":"https://pith.science/api/pith-number/2XP2SLC5I22HV2QKGRKRXOM6W4/graph.json","events_json":"https://pith.science/api/pith-number/2XP2SLC5I22HV2QKGRKRXOM6W4/events.json","paper":"https://pith.science/paper/2XP2SLC5"},"agent_actions":{"view_html":"https://pith.science/pith/2XP2SLC5I22HV2QKGRKRXOM6W4","download_json":"https://pith.science/pith/2XP2SLC5I22HV2QKGRKRXOM6W4.json","view_paper":"https://pith.science/paper/2XP2SLC5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.07317&json=true","fetch_graph":"https://pith.science/api/pith-number/2XP2SLC5I22HV2QKGRKRXOM6W4/graph.json","fetch_events":"https://pith.science/api/pith-number/2XP2SLC5I22HV2QKGRKRXOM6W4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2XP2SLC5I22HV2QKGRKRXOM6W4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2XP2SLC5I22HV2QKGRKRXOM6W4/action/storage_attestation","attest_author":"https://pith.science/pith/2XP2SLC5I22HV2QKGRKRXOM6W4/action/author_attestation","sign_citation":"https://pith.science/pith/2XP2SLC5I22HV2QKGRKRXOM6W4/action/citation_signature","submit_replication":"https://pith.science/pith/2XP2SLC5I22HV2QKGRKRXOM6W4/action/replication_record"}},"created_at":"2026-05-18T00:00:28.296206+00:00","updated_at":"2026-05-18T00:00:28.296206+00:00"}