{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:7KWU6H6RMJY2RFBL6IV3P2U2OT","short_pith_number":"pith:7KWU6H6R","schema_version":"1.0","canonical_sha256":"faad4f1fd16271a8942bf22bb7ea9a74ea39e62bbf3d7c762d962a80c85d96bb","source":{"kind":"arxiv","id":"1804.02209","version":1},"attestation_state":"computed","paper":{"title":"Absolute Continuity of Complex Martingales and of Solutions to Complex Smoothing Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ewa Damek, Sebastian Mentemeier","submitted_at":"2018-04-06T11:27:47Z","abstract_excerpt":"Let $X$ be a $\\mathbb{C}$-valued random variable with the property that $$X \\ \\text{ has the same law as }\\ \\sum_{j\\ge1} T_j X_j$$ where $X_j$ are i.i.d.\\ copies of $X$, which are independent of the (given) $\\mathbb{C}$-valued random variables $ (T_j)_{j\\ge1}$. We provide a simple criterion for the absolute continuity of the law of $X$ that requires, besides the known conditions for the existence of $X$, only finiteness of the first and second moment of $N$ - the number of nonzero weights $T_j$. Our criterion applies in particular to Biggins' martingale with complex parameter."},"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":"1804.02209","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-04-06T11:27:47Z","cross_cats_sorted":[],"title_canon_sha256":"bdf8e728dc7d863e11522efcb5694fbe248a2ae1ca346a4b3bb44d8f321dea38","abstract_canon_sha256":"a2e119c3db05c83170f131e4333ac9fd42503f65c969d151cd170921cb8e4b22"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:05.533146Z","signature_b64":"oqEP4xZ/0aBMgiEnY7F4qt3JzTph3Ca6ySyd5h1X6s8NMoDVlHNnwv7KSEOAGulMCnC5is4p0bs+sZsbdooMCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"faad4f1fd16271a8942bf22bb7ea9a74ea39e62bbf3d7c762d962a80c85d96bb","last_reissued_at":"2026-05-18T00:19:05.532655Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:05.532655Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Absolute Continuity of Complex Martingales and of Solutions to Complex Smoothing Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ewa Damek, Sebastian Mentemeier","submitted_at":"2018-04-06T11:27:47Z","abstract_excerpt":"Let $X$ be a $\\mathbb{C}$-valued random variable with the property that $$X \\ \\text{ has the same law as }\\ \\sum_{j\\ge1} T_j X_j$$ where $X_j$ are i.i.d.\\ copies of $X$, which are independent of the (given) $\\mathbb{C}$-valued random variables $ (T_j)_{j\\ge1}$. We provide a simple criterion for the absolute continuity of the law of $X$ that requires, besides the known conditions for the existence of $X$, only finiteness of the first and second moment of $N$ - the number of nonzero weights $T_j$. Our criterion applies in particular to Biggins' martingale with complex parameter."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.02209","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":"1804.02209","created_at":"2026-05-18T00:19:05.532726+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.02209v1","created_at":"2026-05-18T00:19:05.532726+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.02209","created_at":"2026-05-18T00:19:05.532726+00:00"},{"alias_kind":"pith_short_12","alias_value":"7KWU6H6RMJY2","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_16","alias_value":"7KWU6H6RMJY2RFBL","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_8","alias_value":"7KWU6H6R","created_at":"2026-05-18T12:32:11.075285+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/7KWU6H6RMJY2RFBL6IV3P2U2OT","json":"https://pith.science/pith/7KWU6H6RMJY2RFBL6IV3P2U2OT.json","graph_json":"https://pith.science/api/pith-number/7KWU6H6RMJY2RFBL6IV3P2U2OT/graph.json","events_json":"https://pith.science/api/pith-number/7KWU6H6RMJY2RFBL6IV3P2U2OT/events.json","paper":"https://pith.science/paper/7KWU6H6R"},"agent_actions":{"view_html":"https://pith.science/pith/7KWU6H6RMJY2RFBL6IV3P2U2OT","download_json":"https://pith.science/pith/7KWU6H6RMJY2RFBL6IV3P2U2OT.json","view_paper":"https://pith.science/paper/7KWU6H6R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.02209&json=true","fetch_graph":"https://pith.science/api/pith-number/7KWU6H6RMJY2RFBL6IV3P2U2OT/graph.json","fetch_events":"https://pith.science/api/pith-number/7KWU6H6RMJY2RFBL6IV3P2U2OT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7KWU6H6RMJY2RFBL6IV3P2U2OT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7KWU6H6RMJY2RFBL6IV3P2U2OT/action/storage_attestation","attest_author":"https://pith.science/pith/7KWU6H6RMJY2RFBL6IV3P2U2OT/action/author_attestation","sign_citation":"https://pith.science/pith/7KWU6H6RMJY2RFBL6IV3P2U2OT/action/citation_signature","submit_replication":"https://pith.science/pith/7KWU6H6RMJY2RFBL6IV3P2U2OT/action/replication_record"}},"created_at":"2026-05-18T00:19:05.532726+00:00","updated_at":"2026-05-18T00:19:05.532726+00:00"}