{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:PF6FQCMLKVAALUDUTCW7FEEHE7","short_pith_number":"pith:PF6FQCML","canonical_record":{"source":{"id":"1807.02837","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-07-08T15:13:16Z","cross_cats_sorted":[],"title_canon_sha256":"fdde33f4a66be3ab87c41cab37aa21d8de45a4fb5df66e016aac8eb74ad54d10","abstract_canon_sha256":"a600a75164095e2aeb382e9f410d46c7d3419909779e8de3c54c11c565965484"},"schema_version":"1.0"},"canonical_sha256":"797c58098b554005d07498adf2908727d7bb03510c7c9675fc1eb6d9e015138a","source":{"kind":"arxiv","id":"1807.02837","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.02837","created_at":"2026-05-17T23:40:08Z"},{"alias_kind":"arxiv_version","alias_value":"1807.02837v2","created_at":"2026-05-17T23:40:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.02837","created_at":"2026-05-17T23:40:08Z"},{"alias_kind":"pith_short_12","alias_value":"PF6FQCMLKVAA","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"PF6FQCMLKVAALUDU","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"PF6FQCML","created_at":"2026-05-18T12:32:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:PF6FQCMLKVAALUDUTCW7FEEHE7","target":"record","payload":{"canonical_record":{"source":{"id":"1807.02837","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-07-08T15:13:16Z","cross_cats_sorted":[],"title_canon_sha256":"fdde33f4a66be3ab87c41cab37aa21d8de45a4fb5df66e016aac8eb74ad54d10","abstract_canon_sha256":"a600a75164095e2aeb382e9f410d46c7d3419909779e8de3c54c11c565965484"},"schema_version":"1.0"},"canonical_sha256":"797c58098b554005d07498adf2908727d7bb03510c7c9675fc1eb6d9e015138a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:08.975401Z","signature_b64":"UVyqyjBg89Y8oMpK2S9kZe0PbOeTWkdgzCYoBzx1Lq53y3uCaa+ipMxHT1UfESATLeQf8Y0F3hNs8LkM5SCOCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"797c58098b554005d07498adf2908727d7bb03510c7c9675fc1eb6d9e015138a","last_reissued_at":"2026-05-17T23:40:08.974822Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:08.974822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.02837","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:40:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kiArTeijp78r2WgwYA6GPXeYRU+9P/PPuqlQi/Nai8evFhTpxDAK/zOVsK5KhT7mWCTacbRs+z9T60coCXqRCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T17:57:34.125768Z"},"content_sha256":"f3415107c5fab706f126a5b64a91f2a675dd0ed9f209e872e3c26e09c771886e","schema_version":"1.0","event_id":"sha256:f3415107c5fab706f126a5b64a91f2a675dd0ed9f209e872e3c26e09c771886e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:PF6FQCMLKVAALUDUTCW7FEEHE7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Limit theorems for a class of critical superprocesses with stable branching","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Renming Song, Yan-Xia Ren, Zhenyao Sun","submitted_at":"2018-07-08T15:13:16Z","abstract_excerpt":"We consider a critical superprocess $\\{X;\\mathbf P_\\mu\\}$ with general spatial motion and spatially dependent stable branching mechanism with lowest stable index $\\gamma_0 > 1$. We first show that, under some conditions, $\\mathbf P_{\\mu}(\\|X_t\\|\\neq 0)$ converges to $0$ as $t\\to \\infty$ and is regularly varying with index $(\\gamma_0-1)^{-1}$. Then we show that, for a large class of non-negative testing functions $f$, the distribution of $\\{X_t(f);\\mathbf P_\\mu(\\cdot|\\|X_t\\|\\neq 0)\\}$, after appropriate rescaling, converges weakly to a positive random variable $\\mathbf z^{(\\gamma_0-1)}$ with La"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.02837","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:40:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TMv1gpwl9kAfM4c5mgG9TmwSD20GCumHWHJ2zKkYRP41A0cCssviCFxNK8SpB1rGw3O9cyXZqMOk02kN2Aw7DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T17:57:34.126445Z"},"content_sha256":"6f314ba488abba75237f0a70bfa7ab556e0c68f9cea967ce90a9090d45bfa496","schema_version":"1.0","event_id":"sha256:6f314ba488abba75237f0a70bfa7ab556e0c68f9cea967ce90a9090d45bfa496"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PF6FQCMLKVAALUDUTCW7FEEHE7/bundle.json","state_url":"https://pith.science/pith/PF6FQCMLKVAALUDUTCW7FEEHE7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PF6FQCMLKVAALUDUTCW7FEEHE7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-08T17:57:34Z","links":{"resolver":"https://pith.science/pith/PF6FQCMLKVAALUDUTCW7FEEHE7","bundle":"https://pith.science/pith/PF6FQCMLKVAALUDUTCW7FEEHE7/bundle.json","state":"https://pith.science/pith/PF6FQCMLKVAALUDUTCW7FEEHE7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PF6FQCMLKVAALUDUTCW7FEEHE7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:PF6FQCMLKVAALUDUTCW7FEEHE7","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":"a600a75164095e2aeb382e9f410d46c7d3419909779e8de3c54c11c565965484","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-07-08T15:13:16Z","title_canon_sha256":"fdde33f4a66be3ab87c41cab37aa21d8de45a4fb5df66e016aac8eb74ad54d10"},"schema_version":"1.0","source":{"id":"1807.02837","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.02837","created_at":"2026-05-17T23:40:08Z"},{"alias_kind":"arxiv_version","alias_value":"1807.02837v2","created_at":"2026-05-17T23:40:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.02837","created_at":"2026-05-17T23:40:08Z"},{"alias_kind":"pith_short_12","alias_value":"PF6FQCMLKVAA","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"PF6FQCMLKVAALUDU","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"PF6FQCML","created_at":"2026-05-18T12:32:43Z"}],"graph_snapshots":[{"event_id":"sha256:6f314ba488abba75237f0a70bfa7ab556e0c68f9cea967ce90a9090d45bfa496","target":"graph","created_at":"2026-05-17T23:40:08Z","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 critical superprocess $\\{X;\\mathbf P_\\mu\\}$ with general spatial motion and spatially dependent stable branching mechanism with lowest stable index $\\gamma_0 > 1$. We first show that, under some conditions, $\\mathbf P_{\\mu}(\\|X_t\\|\\neq 0)$ converges to $0$ as $t\\to \\infty$ and is regularly varying with index $(\\gamma_0-1)^{-1}$. Then we show that, for a large class of non-negative testing functions $f$, the distribution of $\\{X_t(f);\\mathbf P_\\mu(\\cdot|\\|X_t\\|\\neq 0)\\}$, after appropriate rescaling, converges weakly to a positive random variable $\\mathbf z^{(\\gamma_0-1)}$ with La","authors_text":"Renming Song, Yan-Xia Ren, Zhenyao Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-07-08T15:13:16Z","title":"Limit theorems for a class of critical superprocesses with stable branching"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.02837","kind":"arxiv","version":2},"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:f3415107c5fab706f126a5b64a91f2a675dd0ed9f209e872e3c26e09c771886e","target":"record","created_at":"2026-05-17T23:40:08Z","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":"a600a75164095e2aeb382e9f410d46c7d3419909779e8de3c54c11c565965484","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-07-08T15:13:16Z","title_canon_sha256":"fdde33f4a66be3ab87c41cab37aa21d8de45a4fb5df66e016aac8eb74ad54d10"},"schema_version":"1.0","source":{"id":"1807.02837","kind":"arxiv","version":2}},"canonical_sha256":"797c58098b554005d07498adf2908727d7bb03510c7c9675fc1eb6d9e015138a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"797c58098b554005d07498adf2908727d7bb03510c7c9675fc1eb6d9e015138a","first_computed_at":"2026-05-17T23:40:08.974822Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:08.974822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UVyqyjBg89Y8oMpK2S9kZe0PbOeTWkdgzCYoBzx1Lq53y3uCaa+ipMxHT1UfESATLeQf8Y0F3hNs8LkM5SCOCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:08.975401Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.02837","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f3415107c5fab706f126a5b64a91f2a675dd0ed9f209e872e3c26e09c771886e","sha256:6f314ba488abba75237f0a70bfa7ab556e0c68f9cea967ce90a9090d45bfa496"],"state_sha256":"b3df904eeaeb64fa2d63270ddff5f3cb5f9b7a665b6a4ac285830ded70c3bded"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WGVdeqm7FSKJEqh+NseH8aJkhoMQKMbFp7YhueIyrmG+TnWA6wBurfCgdoqfxDiF3V+0tHAXrNkO8ZJFPFO7Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T17:57:34.130459Z","bundle_sha256":"105ca068ba61ff8438095c162855dff925855540565fedc4bb9fba086ea4235f"}}