{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:DFBHGOCMRTGZOYHW5TBGPNJXBU","short_pith_number":"pith:DFBHGOCM","schema_version":"1.0","canonical_sha256":"194273384c8ccd9760f6ecc267b5370d03bfde4e17ba3f0122f19c62561c1819","source":{"kind":"arxiv","id":"1602.08181","version":2},"attestation_state":"computed","paper":{"title":"Ratios and Cauchy Distribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR","stat.TH"],"primary_cat":"math.ST","authors_text":"Natesh S. Pillai","submitted_at":"2016-02-26T02:47:13Z","abstract_excerpt":"It is well known that the ratio of two independent standard Gaussian random variables follows a Cauchy distribution. Any convex combination of independent standard Cauchy random variables also follows a Cauchy distribution. In a recent joint work, the author proved a surprising multivariate generalization of the above facts. Fix $m > 1$ and let $\\Sigma$ be a $m\\times m$ positive semi-definite matrix. Let $X,Y \\sim \\mathrm{N}(0,\\Sigma)$ be independent vectors. Let $\\vec{w}=(w_1, \\dots, w_m)$ be a vector of non-negative numbers with $\\sum_{j=1}^m w_j = 1.$ The author proved recently that the ran"},"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":"1602.08181","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2016-02-26T02:47:13Z","cross_cats_sorted":["math.PR","stat.TH"],"title_canon_sha256":"704c5b0e6910ed1cdba5dbd095d92a61b1fbb7f74fb5994f180bdff196198a72","abstract_canon_sha256":"62c2f628ddf7f75a605d7164891b00aef45bd32f9be514e448ee4675f053b54c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:40.202426Z","signature_b64":"C1J6dYd13WYAGKd1Y/nawwX7gWak0t1J35YAByctpjbhmJZQWyUm2Ou7fkr45VuLfcRTtvuIikNjs/4GKy5yBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"194273384c8ccd9760f6ecc267b5370d03bfde4e17ba3f0122f19c62561c1819","last_reissued_at":"2026-05-18T01:19:40.201725Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:40.201725Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ratios and Cauchy Distribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR","stat.TH"],"primary_cat":"math.ST","authors_text":"Natesh S. Pillai","submitted_at":"2016-02-26T02:47:13Z","abstract_excerpt":"It is well known that the ratio of two independent standard Gaussian random variables follows a Cauchy distribution. Any convex combination of independent standard Cauchy random variables also follows a Cauchy distribution. In a recent joint work, the author proved a surprising multivariate generalization of the above facts. Fix $m > 1$ and let $\\Sigma$ be a $m\\times m$ positive semi-definite matrix. Let $X,Y \\sim \\mathrm{N}(0,\\Sigma)$ be independent vectors. Let $\\vec{w}=(w_1, \\dots, w_m)$ be a vector of non-negative numbers with $\\sum_{j=1}^m w_j = 1.$ The author proved recently that the ran"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.08181","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1602.08181","created_at":"2026-05-18T01:19:40.201841+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.08181v2","created_at":"2026-05-18T01:19:40.201841+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.08181","created_at":"2026-05-18T01:19:40.201841+00:00"},{"alias_kind":"pith_short_12","alias_value":"DFBHGOCMRTGZ","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_16","alias_value":"DFBHGOCMRTGZOYHW","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_8","alias_value":"DFBHGOCM","created_at":"2026-05-18T12:30:12.583610+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2411.12555","citing_title":"Multivariate and Online Transfer Learning with Uncertainty Quantification","ref_index":2992,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DFBHGOCMRTGZOYHW5TBGPNJXBU","json":"https://pith.science/pith/DFBHGOCMRTGZOYHW5TBGPNJXBU.json","graph_json":"https://pith.science/api/pith-number/DFBHGOCMRTGZOYHW5TBGPNJXBU/graph.json","events_json":"https://pith.science/api/pith-number/DFBHGOCMRTGZOYHW5TBGPNJXBU/events.json","paper":"https://pith.science/paper/DFBHGOCM"},"agent_actions":{"view_html":"https://pith.science/pith/DFBHGOCMRTGZOYHW5TBGPNJXBU","download_json":"https://pith.science/pith/DFBHGOCMRTGZOYHW5TBGPNJXBU.json","view_paper":"https://pith.science/paper/DFBHGOCM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.08181&json=true","fetch_graph":"https://pith.science/api/pith-number/DFBHGOCMRTGZOYHW5TBGPNJXBU/graph.json","fetch_events":"https://pith.science/api/pith-number/DFBHGOCMRTGZOYHW5TBGPNJXBU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DFBHGOCMRTGZOYHW5TBGPNJXBU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DFBHGOCMRTGZOYHW5TBGPNJXBU/action/storage_attestation","attest_author":"https://pith.science/pith/DFBHGOCMRTGZOYHW5TBGPNJXBU/action/author_attestation","sign_citation":"https://pith.science/pith/DFBHGOCMRTGZOYHW5TBGPNJXBU/action/citation_signature","submit_replication":"https://pith.science/pith/DFBHGOCMRTGZOYHW5TBGPNJXBU/action/replication_record"}},"created_at":"2026-05-18T01:19:40.201841+00:00","updated_at":"2026-05-18T01:19:40.201841+00:00"}