{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:WZD7NGYPZOMO7CNIFUYIF75Q6O","short_pith_number":"pith:WZD7NGYP","canonical_record":{"source":{"id":"1210.7732","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-10-29T17:10:32Z","cross_cats_sorted":[],"title_canon_sha256":"6b198f1884e8745bdbf98d60bf69b1b267d8edb566e003a8de5e28847a1bfdec","abstract_canon_sha256":"72b99c996ddb7dcd47cdd5087cfd8ffaddc24a4c0a67c60d140ac56925d7050d"},"schema_version":"1.0"},"canonical_sha256":"b647f69b0fcb98ef89a82d3082ffb0f3a3a57b23c24b0bea4150cfff29bda57c","source":{"kind":"arxiv","id":"1210.7732","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.7732","created_at":"2026-05-18T03:41:59Z"},{"alias_kind":"arxiv_version","alias_value":"1210.7732v1","created_at":"2026-05-18T03:41:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.7732","created_at":"2026-05-18T03:41:59Z"},{"alias_kind":"pith_short_12","alias_value":"WZD7NGYPZOMO","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"WZD7NGYPZOMO7CNI","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"WZD7NGYP","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:WZD7NGYPZOMO7CNIFUYIF75Q6O","target":"record","payload":{"canonical_record":{"source":{"id":"1210.7732","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-10-29T17:10:32Z","cross_cats_sorted":[],"title_canon_sha256":"6b198f1884e8745bdbf98d60bf69b1b267d8edb566e003a8de5e28847a1bfdec","abstract_canon_sha256":"72b99c996ddb7dcd47cdd5087cfd8ffaddc24a4c0a67c60d140ac56925d7050d"},"schema_version":"1.0"},"canonical_sha256":"b647f69b0fcb98ef89a82d3082ffb0f3a3a57b23c24b0bea4150cfff29bda57c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:59.491143Z","signature_b64":"cHoG44fzNMqF+db+mMIsFbI5cRug0NGrOgV4r42n77IgwXE3lss8tJa/mhHtLZlRN1OF3xckkGMzMfQrJYIaAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b647f69b0fcb98ef89a82d3082ffb0f3a3a57b23c24b0bea4150cfff29bda57c","last_reissued_at":"2026-05-18T03:41:59.490428Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:59.490428Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.7732","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:41:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V5f0LXpnPCVcdr8l1N7q8Js6r6KHws/HUFoiA0Am7V1dexpk1bwegHHMj4PHXEmkyz8jgX8h5iGhs/OR5pYgAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:47:42.336042Z"},"content_sha256":"bd8196eb7e74a3fe5bc88a1ff22bee959680c04339ffecc4f039d32c5838a380","schema_version":"1.0","event_id":"sha256:bd8196eb7e74a3fe5bc88a1ff22bee959680c04339ffecc4f039d32c5838a380"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:WZD7NGYPZOMO7CNIFUYIF75Q6O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Linear stochastic equations in the critical case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dariusz Buraczewski, Konrad Kolesko","submitted_at":"2012-10-29T17:10:32Z","abstract_excerpt":"We consider solutions of the stochastic equation $X \\stackrel{d}= \\sum_{i=1}^N A_iX_i + B$, where $N$ is a random natural number, $B$ and $A_i$ are random positive numbers and $X_i$ are independent copies of $X$, which are independent also of $N,B,A_i$. Properties of solutions of this equation are mainly coded in the function $m(s)=\\mathbb{E}\\big[\\sum_{i=1}^N A_i^s \\big]$. In this paper we study the critical case when the function $m$ is tangent to the line $y=1$. Then, under a number of further assumptions, we prove existence of solutions and describe their asymptotic behavior."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.7732","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:41:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1vxvcocQOf7wo0EFCzOeZ7YZl0S6A0DQETogBjjwVIjR81iXJX/vw6KECD/TkmXlTVC6B7sCgzPe1RuCvZkODQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:47:42.336742Z"},"content_sha256":"69079f7ffe5ebcb03d64629e85e0059222d8846eaaf3897ee5c3275186a9c612","schema_version":"1.0","event_id":"sha256:69079f7ffe5ebcb03d64629e85e0059222d8846eaaf3897ee5c3275186a9c612"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WZD7NGYPZOMO7CNIFUYIF75Q6O/bundle.json","state_url":"https://pith.science/pith/WZD7NGYPZOMO7CNIFUYIF75Q6O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WZD7NGYPZOMO7CNIFUYIF75Q6O/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":"2026-06-06T20:47:42Z","links":{"resolver":"https://pith.science/pith/WZD7NGYPZOMO7CNIFUYIF75Q6O","bundle":"https://pith.science/pith/WZD7NGYPZOMO7CNIFUYIF75Q6O/bundle.json","state":"https://pith.science/pith/WZD7NGYPZOMO7CNIFUYIF75Q6O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WZD7NGYPZOMO7CNIFUYIF75Q6O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:WZD7NGYPZOMO7CNIFUYIF75Q6O","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":"72b99c996ddb7dcd47cdd5087cfd8ffaddc24a4c0a67c60d140ac56925d7050d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-10-29T17:10:32Z","title_canon_sha256":"6b198f1884e8745bdbf98d60bf69b1b267d8edb566e003a8de5e28847a1bfdec"},"schema_version":"1.0","source":{"id":"1210.7732","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.7732","created_at":"2026-05-18T03:41:59Z"},{"alias_kind":"arxiv_version","alias_value":"1210.7732v1","created_at":"2026-05-18T03:41:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.7732","created_at":"2026-05-18T03:41:59Z"},{"alias_kind":"pith_short_12","alias_value":"WZD7NGYPZOMO","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"WZD7NGYPZOMO7CNI","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"WZD7NGYP","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:69079f7ffe5ebcb03d64629e85e0059222d8846eaaf3897ee5c3275186a9c612","target":"graph","created_at":"2026-05-18T03:41:59Z","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 solutions of the stochastic equation $X \\stackrel{d}= \\sum_{i=1}^N A_iX_i + B$, where $N$ is a random natural number, $B$ and $A_i$ are random positive numbers and $X_i$ are independent copies of $X$, which are independent also of $N,B,A_i$. Properties of solutions of this equation are mainly coded in the function $m(s)=\\mathbb{E}\\big[\\sum_{i=1}^N A_i^s \\big]$. In this paper we study the critical case when the function $m$ is tangent to the line $y=1$. Then, under a number of further assumptions, we prove existence of solutions and describe their asymptotic behavior.","authors_text":"Dariusz Buraczewski, Konrad Kolesko","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-10-29T17:10:32Z","title":"Linear stochastic equations in the critical case"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.7732","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:bd8196eb7e74a3fe5bc88a1ff22bee959680c04339ffecc4f039d32c5838a380","target":"record","created_at":"2026-05-18T03:41:59Z","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":"72b99c996ddb7dcd47cdd5087cfd8ffaddc24a4c0a67c60d140ac56925d7050d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-10-29T17:10:32Z","title_canon_sha256":"6b198f1884e8745bdbf98d60bf69b1b267d8edb566e003a8de5e28847a1bfdec"},"schema_version":"1.0","source":{"id":"1210.7732","kind":"arxiv","version":1}},"canonical_sha256":"b647f69b0fcb98ef89a82d3082ffb0f3a3a57b23c24b0bea4150cfff29bda57c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b647f69b0fcb98ef89a82d3082ffb0f3a3a57b23c24b0bea4150cfff29bda57c","first_computed_at":"2026-05-18T03:41:59.490428Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:59.490428Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cHoG44fzNMqF+db+mMIsFbI5cRug0NGrOgV4r42n77IgwXE3lss8tJa/mhHtLZlRN1OF3xckkGMzMfQrJYIaAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:59.491143Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.7732","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bd8196eb7e74a3fe5bc88a1ff22bee959680c04339ffecc4f039d32c5838a380","sha256:69079f7ffe5ebcb03d64629e85e0059222d8846eaaf3897ee5c3275186a9c612"],"state_sha256":"387d8c77988da6a8322eeb9b516382c1825ac9ce736f277967594bf6271c0528"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VssfnpD5uVnb1jDdFHy2eS5rHs/W5hX29/2DWDWTijMF+iXldfNCQAMb+4d/6dHwhiBDaghDzTwufOrvFZqLBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T20:47:42.340174Z","bundle_sha256":"42cab19829ea7c95d13d6ec8c54785db13317978d2e1b5f48f30704c8a189305"}}