{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:WQ765XEX6IOH3IX4XVI6E253KF","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":"e2dfb6a9bb3d8cf2e1d1c9d18d348bd903d173cde736090b1d86daf7fcfa4bc2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-26T20:26:18Z","title_canon_sha256":"7929e55aef21daf4bb956ebbf0e7240a081e5f76c309d76d41301472dea4c62b"},"schema_version":"1.0","source":{"id":"1508.06653","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.06653","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"arxiv_version","alias_value":"1508.06653v1","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.06653","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"pith_short_12","alias_value":"WQ765XEX6IOH","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WQ765XEX6IOH3IX4","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WQ765XEX","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:f056e6073b44ac4f8fbf46dc716ed0da2660c139e46cd346213a155376cf8c46","target":"graph","created_at":"2026-05-18T01:34:41Z","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":"We consider a Galton-Watson process $\\mathbf{Z}% (n)=(Z_{1}(n),Z_{2}(n))$ with two types of particles. Particles of type 2 may produce offspring of both types while particles of type 1 may produce particles of their own type only. Let $Z_{i}(m,n)$ be the number of particles of type $i$ at time $m<n$ having offspring at time $n$. Assuming that the process is critical and that the variance of the offspring distribution may be infinite we describe the asymptotic behavior, as $m,n\\rightarrow \\infty $ of the law of $\\mathbf{Z}(m,n)=(Z_1(m,n),Z_2(m,n))$ given $\\mathbf{Z}(n)\\neq \\mathbf{0}$. We find ","authors_text":"Charline Smadi, Vladimir A. Vatutin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-26T20:26:18Z","title":"Reduced two-type decomposable critical branching processes with possibly infinite variance"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06653","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":"sha256:6077f44d86ec1ee5e3a2c87f48f7715cde72720355db39d2e9acbd16278ca046","target":"record","created_at":"2026-05-18T01:34:41Z","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":"e2dfb6a9bb3d8cf2e1d1c9d18d348bd903d173cde736090b1d86daf7fcfa4bc2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-26T20:26:18Z","title_canon_sha256":"7929e55aef21daf4bb956ebbf0e7240a081e5f76c309d76d41301472dea4c62b"},"schema_version":"1.0","source":{"id":"1508.06653","kind":"arxiv","version":1}},"canonical_sha256":"b43feedc97f21c7da2fcbd51e26bbb5163109ae19a3824869a65af4e9396b94c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b43feedc97f21c7da2fcbd51e26bbb5163109ae19a3824869a65af4e9396b94c","first_computed_at":"2026-05-18T01:34:41.414329Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:41.414329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3JRVOjHsy4sO/EBsgx+it3JcTUSvxVCJG1nsmTEkdQbowNhfCa+hzpLd7UKXFlHfRi5/qyuOyrTJQVdBV/efAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:41.415161Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.06653","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6077f44d86ec1ee5e3a2c87f48f7715cde72720355db39d2e9acbd16278ca046","sha256:f056e6073b44ac4f8fbf46dc716ed0da2660c139e46cd346213a155376cf8c46"],"state_sha256":"4c5b98156c67d3a61ca5c8d6a70ea113f12456a094045d75214c3dfc8fca0503"}