{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:IRCMWPAQYZQQR5GIVUP2MIRVXR","short_pith_number":"pith:IRCMWPAQ","schema_version":"1.0","canonical_sha256":"4444cb3c10c66108f4c8ad1fa62235bc6d770e802f2301e5c3db0030042df482","source":{"kind":"arxiv","id":"1207.6874","version":1},"attestation_state":"computed","paper":{"title":"Heavy tailed branching process with immigration","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Bojan Basrak, Rafa{\\l} Kulik, Zbigniew Palmowski","submitted_at":"2012-07-30T09:32:04Z","abstract_excerpt":"In this paper we analyze a branching process with immigration defined recursively by $X_t=\\theta_t\\circ X_{t-1}+B_t$ for a sequence $(B_t)$ of i.i.d. random variables and random mappings $ \\theta_t\\circ x:=\\theta_t(x)=\\sum_{i=1}^xA_i^{(t)}, $ with $(A_i^{(t)})_{i\\in \\mathbb{N}_0}$ being a sequence of $\\mathbb{N}_0$-valued i.i.d. random variables independent of $B_t$. We assume that one of generic variables $A$ and $B$ has a regularly varying tail distribution. We identify the tail behaviour of the distribution of the stationary solution $X_t$. We also prove CLT for the partial sums that could "},"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":"1207.6874","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-30T09:32:04Z","cross_cats_sorted":[],"title_canon_sha256":"09829156fe993e9ea6749a14a67924675ad1f56fbe8b930d03fb4c129e65169d","abstract_canon_sha256":"dd72db7f03469d2fd7f679202b3c7a9ca8495918ec8740e86cc65b33e46ab420"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:48.372944Z","signature_b64":"TQ8FcN7bq/iFkN6VYy55SErAWw6H+2uQbJa/pryGFLU1MjP9r+y6vahQ0QXfOetUM0aYxgCJfjkh0XqbttZ5DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4444cb3c10c66108f4c8ad1fa62235bc6d770e802f2301e5c3db0030042df482","last_reissued_at":"2026-05-18T03:49:48.372070Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:48.372070Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Heavy tailed branching process with immigration","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Bojan Basrak, Rafa{\\l} Kulik, Zbigniew Palmowski","submitted_at":"2012-07-30T09:32:04Z","abstract_excerpt":"In this paper we analyze a branching process with immigration defined recursively by $X_t=\\theta_t\\circ X_{t-1}+B_t$ for a sequence $(B_t)$ of i.i.d. random variables and random mappings $ \\theta_t\\circ x:=\\theta_t(x)=\\sum_{i=1}^xA_i^{(t)}, $ with $(A_i^{(t)})_{i\\in \\mathbb{N}_0}$ being a sequence of $\\mathbb{N}_0$-valued i.i.d. random variables independent of $B_t$. We assume that one of generic variables $A$ and $B$ has a regularly varying tail distribution. We identify the tail behaviour of the distribution of the stationary solution $X_t$. We also prove CLT for the partial sums that could "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6874","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":"1207.6874","created_at":"2026-05-18T03:49:48.372221+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.6874v1","created_at":"2026-05-18T03:49:48.372221+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.6874","created_at":"2026-05-18T03:49:48.372221+00:00"},{"alias_kind":"pith_short_12","alias_value":"IRCMWPAQYZQQ","created_at":"2026-05-18T12:27:09.501522+00:00"},{"alias_kind":"pith_short_16","alias_value":"IRCMWPAQYZQQR5GI","created_at":"2026-05-18T12:27:09.501522+00:00"},{"alias_kind":"pith_short_8","alias_value":"IRCMWPAQ","created_at":"2026-05-18T12:27:09.501522+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/IRCMWPAQYZQQR5GIVUP2MIRVXR","json":"https://pith.science/pith/IRCMWPAQYZQQR5GIVUP2MIRVXR.json","graph_json":"https://pith.science/api/pith-number/IRCMWPAQYZQQR5GIVUP2MIRVXR/graph.json","events_json":"https://pith.science/api/pith-number/IRCMWPAQYZQQR5GIVUP2MIRVXR/events.json","paper":"https://pith.science/paper/IRCMWPAQ"},"agent_actions":{"view_html":"https://pith.science/pith/IRCMWPAQYZQQR5GIVUP2MIRVXR","download_json":"https://pith.science/pith/IRCMWPAQYZQQR5GIVUP2MIRVXR.json","view_paper":"https://pith.science/paper/IRCMWPAQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.6874&json=true","fetch_graph":"https://pith.science/api/pith-number/IRCMWPAQYZQQR5GIVUP2MIRVXR/graph.json","fetch_events":"https://pith.science/api/pith-number/IRCMWPAQYZQQR5GIVUP2MIRVXR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IRCMWPAQYZQQR5GIVUP2MIRVXR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IRCMWPAQYZQQR5GIVUP2MIRVXR/action/storage_attestation","attest_author":"https://pith.science/pith/IRCMWPAQYZQQR5GIVUP2MIRVXR/action/author_attestation","sign_citation":"https://pith.science/pith/IRCMWPAQYZQQR5GIVUP2MIRVXR/action/citation_signature","submit_replication":"https://pith.science/pith/IRCMWPAQYZQQR5GIVUP2MIRVXR/action/replication_record"}},"created_at":"2026-05-18T03:49:48.372221+00:00","updated_at":"2026-05-18T03:49:48.372221+00:00"}