{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:XRUSBC4RXEUTQMS5G74ODTN2WV","short_pith_number":"pith:XRUSBC4R","schema_version":"1.0","canonical_sha256":"bc69208b91b92938325d37f8e1cdbab55401a4ba3b1502632dc13ad138237359","source":{"kind":"arxiv","id":"1706.09718","version":4},"attestation_state":"computed","paper":{"title":"Independence characterization for Wishart and Kummer random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Agnieszka Piliszek, Bartosz Ko{\\l}odziejek","submitted_at":"2017-06-29T12:35:35Z","abstract_excerpt":"We generalize the following univariate characterization of the Kummer and Gamma distributions to the cone of symmetric positive definite matrices: let $X$ and $Y$ be independent, non-degenerate random variables valued in $(0, \\infty)$, then $U= Y/(1+X)$ and $V = X(1+U)$ are independent if and only if $X$ follows the Kummer distribution and $Y$ follows the the Gamma distribution with appropriate parameters. We solve a related functional equation in the cone of symmetric positive definite matrices, which is our first main result and apply its solution to prove the characterization of Wishart and"},"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":"1706.09718","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-06-29T12:35:35Z","cross_cats_sorted":[],"title_canon_sha256":"1770107513b5e301eebbb4979870e541dd4754bc09f271163afd29a45a310c8e","abstract_canon_sha256":"aefe1a9c978b7f7fddbb4617746cf51ef713e10a2922a005727778c1b9025687"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:59.288811Z","signature_b64":"Y5t4/bjdWVDQH92g2L8S0nNMjdoIO5pozy8PyZHx5y8iF22iy9VXKG/0sXm7QYdqAR1Be5BYZ1iRbNd0/TYUAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc69208b91b92938325d37f8e1cdbab55401a4ba3b1502632dc13ad138237359","last_reissued_at":"2026-05-18T00:15:59.288294Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:59.288294Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Independence characterization for Wishart and Kummer random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Agnieszka Piliszek, Bartosz Ko{\\l}odziejek","submitted_at":"2017-06-29T12:35:35Z","abstract_excerpt":"We generalize the following univariate characterization of the Kummer and Gamma distributions to the cone of symmetric positive definite matrices: let $X$ and $Y$ be independent, non-degenerate random variables valued in $(0, \\infty)$, then $U= Y/(1+X)$ and $V = X(1+U)$ are independent if and only if $X$ follows the Kummer distribution and $Y$ follows the the Gamma distribution with appropriate parameters. We solve a related functional equation in the cone of symmetric positive definite matrices, which is our first main result and apply its solution to prove the characterization of Wishart and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.09718","kind":"arxiv","version":4},"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":"1706.09718","created_at":"2026-05-18T00:15:59.288381+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.09718v4","created_at":"2026-05-18T00:15:59.288381+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.09718","created_at":"2026-05-18T00:15:59.288381+00:00"},{"alias_kind":"pith_short_12","alias_value":"XRUSBC4RXEUT","created_at":"2026-05-18T12:31:56.362134+00:00"},{"alias_kind":"pith_short_16","alias_value":"XRUSBC4RXEUTQMS5","created_at":"2026-05-18T12:31:56.362134+00:00"},{"alias_kind":"pith_short_8","alias_value":"XRUSBC4R","created_at":"2026-05-18T12:31:56.362134+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/XRUSBC4RXEUTQMS5G74ODTN2WV","json":"https://pith.science/pith/XRUSBC4RXEUTQMS5G74ODTN2WV.json","graph_json":"https://pith.science/api/pith-number/XRUSBC4RXEUTQMS5G74ODTN2WV/graph.json","events_json":"https://pith.science/api/pith-number/XRUSBC4RXEUTQMS5G74ODTN2WV/events.json","paper":"https://pith.science/paper/XRUSBC4R"},"agent_actions":{"view_html":"https://pith.science/pith/XRUSBC4RXEUTQMS5G74ODTN2WV","download_json":"https://pith.science/pith/XRUSBC4RXEUTQMS5G74ODTN2WV.json","view_paper":"https://pith.science/paper/XRUSBC4R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.09718&json=true","fetch_graph":"https://pith.science/api/pith-number/XRUSBC4RXEUTQMS5G74ODTN2WV/graph.json","fetch_events":"https://pith.science/api/pith-number/XRUSBC4RXEUTQMS5G74ODTN2WV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XRUSBC4RXEUTQMS5G74ODTN2WV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XRUSBC4RXEUTQMS5G74ODTN2WV/action/storage_attestation","attest_author":"https://pith.science/pith/XRUSBC4RXEUTQMS5G74ODTN2WV/action/author_attestation","sign_citation":"https://pith.science/pith/XRUSBC4RXEUTQMS5G74ODTN2WV/action/citation_signature","submit_replication":"https://pith.science/pith/XRUSBC4RXEUTQMS5G74ODTN2WV/action/replication_record"}},"created_at":"2026-05-18T00:15:59.288381+00:00","updated_at":"2026-05-18T00:15:59.288381+00:00"}