{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:75ZYBJ2USWMNYQ2Q7UREW4SXGQ","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":"8fe77447281e0d9cef67d059e2bb44643f8a27886750dc8e7665e4c9c75f7ede","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-12-24T08:59:06Z","title_canon_sha256":"b8ebb91aaba1a73e0065aa4b019e12e8e7020e627634655adb234e0a6f902b5f"},"schema_version":"1.0","source":{"id":"1712.08892","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.08892","created_at":"2026-05-18T00:27:14Z"},{"alias_kind":"arxiv_version","alias_value":"1712.08892v1","created_at":"2026-05-18T00:27:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.08892","created_at":"2026-05-18T00:27:14Z"},{"alias_kind":"pith_short_12","alias_value":"75ZYBJ2USWMN","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_16","alias_value":"75ZYBJ2USWMNYQ2Q","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_8","alias_value":"75ZYBJ2U","created_at":"2026-05-18T12:31:03Z"}],"graph_snapshots":[{"event_id":"sha256:f748ccf73e94997e756b077285e2dc7d004e9097825dc6ce7ce05787c08a42bb","target":"graph","created_at":"2026-05-18T00:27:14Z","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":"In this paper, we investigate the asymptotic behaviors of the critical branching process with immigration $\\{Z_n, n\\ge 0\\}$. First we get some estimation for the probability generating function of $Z_n$. Based on it, we get a large deviation for $Z_{n+1}/Z_n$. Lower and upper deviations for $Z_n$ are also studied. As a by-product, an upper deviation for $\\max_{1\\le i\\le n} Z_i$ is obtained.","authors_text":"Doudou Li, Mei Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-12-24T08:59:06Z","title":"Asymptotic Behaviors for Critical Branching Processes with Immigration"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08892","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:d79fffb231f43fe2b03b652834e3b7d7918719be36c2f76a47bb751afe911f7f","target":"record","created_at":"2026-05-18T00:27:14Z","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":"8fe77447281e0d9cef67d059e2bb44643f8a27886750dc8e7665e4c9c75f7ede","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-12-24T08:59:06Z","title_canon_sha256":"b8ebb91aaba1a73e0065aa4b019e12e8e7020e627634655adb234e0a6f902b5f"},"schema_version":"1.0","source":{"id":"1712.08892","kind":"arxiv","version":1}},"canonical_sha256":"ff7380a7549598dc4350fd224b725734071e2181c9d928adb368e64756b02cb4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ff7380a7549598dc4350fd224b725734071e2181c9d928adb368e64756b02cb4","first_computed_at":"2026-05-18T00:27:14.895427Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:27:14.895427Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HtAJFiNhNTZmlvAy8Ae96zKyvbz3zajkWoZTwcRzCsYZM+XrS5MOeDSrlZROxqgZ5/NI+Mx2o4X4urUAGGrGBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:27:14.895926Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.08892","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d79fffb231f43fe2b03b652834e3b7d7918719be36c2f76a47bb751afe911f7f","sha256:f748ccf73e94997e756b077285e2dc7d004e9097825dc6ce7ce05787c08a42bb"],"state_sha256":"0df5503754e28dee140ae65d641072ed5782894c02ec91233c3495be02879561"}