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We assume that we are in Schr\\\"oder case with $EZ_1\\log Z_1<\\infty$ and $X_1$ is in the domain of attraction of an $\\alpha$-stable law with $0<\\alpha<2$. As by-products, when $Z_1$ is sub-exponentially distributed, we further obtain the convergence rates of $ \\frac{Z_{n+1}}{Z_n}$ to $m$ as $n\\rightarrow\\infty$."},"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":"1502.01433","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-05T04:50:50Z","cross_cats_sorted":[],"title_canon_sha256":"cb45b738d115092f9927decae7b11b26616478990f0d22f70ffbfc9909e536f6","abstract_canon_sha256":"71adf8daa45f67679b204dfa29106fc0ade84677a32d940388b4b79dc440d098"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:37.901944Z","signature_b64":"xJ/oWjgbDLbUtK1hT9++FNJYbMnESGoRi16NKwtPEk/3/J8X14jgiw6IExLkb83Nla+44bMILni9sQbJojb5BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea74783b24a8003ae39757d78b3d75250df8e551b8eb47f2b31b36350ca844d1","last_reissued_at":"2026-05-18T01:34:37.901043Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:37.901043Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On large deviation rates for sums associated with Galton-Watson processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hui He","submitted_at":"2015-02-05T04:50:50Z","abstract_excerpt":"Given a super-critical Galton-Watson process $\\{Z_n\\}$ and a positive sequence $\\{\\epsilon_n\\}$, we study the limiting behaviors of $P(S_{Z_n}/Z_n\\geq\\epsilon_n)$ and $P(S_{Z_n}/m^n\\geq\\epsilon_n) $ with sums $S_{n}$ of i.i.d. random variables $X_i$ and $m=E[Z_1]$. We assume that we are in Schr\\\"oder case with $EZ_1\\log Z_1<\\infty$ and $X_1$ is in the domain of attraction of an $\\alpha$-stable law with $0<\\alpha<2$. As by-products, when $Z_1$ is sub-exponentially distributed, we further obtain the convergence rates of $ \\frac{Z_{n+1}}{Z_n}$ to $m$ as $n\\rightarrow\\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01433","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":"1502.01433","created_at":"2026-05-18T01:34:37.901190+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.01433v2","created_at":"2026-05-18T01:34:37.901190+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01433","created_at":"2026-05-18T01:34:37.901190+00:00"},{"alias_kind":"pith_short_12","alias_value":"5J2HQOZEVAAD","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"5J2HQOZEVAADVY4X","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"5J2HQOZE","created_at":"2026-05-18T12:29:05.191682+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/5J2HQOZEVAADVY4XK7LYWPLVEU","json":"https://pith.science/pith/5J2HQOZEVAADVY4XK7LYWPLVEU.json","graph_json":"https://pith.science/api/pith-number/5J2HQOZEVAADVY4XK7LYWPLVEU/graph.json","events_json":"https://pith.science/api/pith-number/5J2HQOZEVAADVY4XK7LYWPLVEU/events.json","paper":"https://pith.science/paper/5J2HQOZE"},"agent_actions":{"view_html":"https://pith.science/pith/5J2HQOZEVAADVY4XK7LYWPLVEU","download_json":"https://pith.science/pith/5J2HQOZEVAADVY4XK7LYWPLVEU.json","view_paper":"https://pith.science/paper/5J2HQOZE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.01433&json=true","fetch_graph":"https://pith.science/api/pith-number/5J2HQOZEVAADVY4XK7LYWPLVEU/graph.json","fetch_events":"https://pith.science/api/pith-number/5J2HQOZEVAADVY4XK7LYWPLVEU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5J2HQOZEVAADVY4XK7LYWPLVEU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5J2HQOZEVAADVY4XK7LYWPLVEU/action/storage_attestation","attest_author":"https://pith.science/pith/5J2HQOZEVAADVY4XK7LYWPLVEU/action/author_attestation","sign_citation":"https://pith.science/pith/5J2HQOZEVAADVY4XK7LYWPLVEU/action/citation_signature","submit_replication":"https://pith.science/pith/5J2HQOZEVAADVY4XK7LYWPLVEU/action/replication_record"}},"created_at":"2026-05-18T01:34:37.901190+00:00","updated_at":"2026-05-18T01:34:37.901190+00:00"}