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We show that the density of the power transformation $T_\\al^\\beta$ is hyperbolically completely monotone in the sense of Thorin and Bondesson if and only if $\\al\\le1/2$ and $|\\beta|\\ge 1/(1-\\al).$ This clarifies a conjecture of Bondesson (1992) on positive stable densities."},"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":"1402.1059","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-02-05T14:38:23Z","cross_cats_sorted":[],"title_canon_sha256":"b7a297568dafe35199663186a2a389a1b9fdb01e5fc1a07a61d376c9551af082","abstract_canon_sha256":"b2c59f47892d87089b29591f0b6938142c801208699652c3bc00387c69f07831"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:01.839342Z","signature_b64":"m0LVVeP9s8gEQd30tS7yhli2wS7zfaCDMTbuyi+3gQLufNBAoOWAYCpyS65GWR+31+c0dO4qu7ta7G16bJ1iCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c98ae696bbe00c3790df459622cf193d8f8f430a4e57bc0624ba0c2f589fd4b","last_reissued_at":"2026-05-18T03:00:01.838607Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:01.838607Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"HCM Property and the Half-Cauchy Distribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Pierre Bosch (LPP)","submitted_at":"2014-02-05T14:38:23Z","abstract_excerpt":"Let $Z_\\al$ be a positive $\\alpha$-stable random variable and $T_\\al=(Z_\\al/\\tilde Z_\\al)^\\al,$ with independents components in the quotient. It is known that $T_\\al$ is distributed as the positive branch of a Cauchy random variable with drift. We show that the density of the power transformation $T_\\al^\\beta$ is hyperbolically completely monotone in the sense of Thorin and Bondesson if and only if $\\al\\le1/2$ and $|\\beta|\\ge 1/(1-\\al).$ This clarifies a conjecture of Bondesson (1992) on positive stable densities."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.1059","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":"1402.1059","created_at":"2026-05-18T03:00:01.838728+00:00"},{"alias_kind":"arxiv_version","alias_value":"1402.1059v1","created_at":"2026-05-18T03:00:01.838728+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.1059","created_at":"2026-05-18T03:00:01.838728+00:00"},{"alias_kind":"pith_short_12","alias_value":"TSMK42LLXYAM","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_16","alias_value":"TSMK42LLXYAMG6IN","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_8","alias_value":"TSMK42LL","created_at":"2026-05-18T12:28:52.271510+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/TSMK42LLXYAMG6IN6RMWELHRSP","json":"https://pith.science/pith/TSMK42LLXYAMG6IN6RMWELHRSP.json","graph_json":"https://pith.science/api/pith-number/TSMK42LLXYAMG6IN6RMWELHRSP/graph.json","events_json":"https://pith.science/api/pith-number/TSMK42LLXYAMG6IN6RMWELHRSP/events.json","paper":"https://pith.science/paper/TSMK42LL"},"agent_actions":{"view_html":"https://pith.science/pith/TSMK42LLXYAMG6IN6RMWELHRSP","download_json":"https://pith.science/pith/TSMK42LLXYAMG6IN6RMWELHRSP.json","view_paper":"https://pith.science/paper/TSMK42LL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1402.1059&json=true","fetch_graph":"https://pith.science/api/pith-number/TSMK42LLXYAMG6IN6RMWELHRSP/graph.json","fetch_events":"https://pith.science/api/pith-number/TSMK42LLXYAMG6IN6RMWELHRSP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TSMK42LLXYAMG6IN6RMWELHRSP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TSMK42LLXYAMG6IN6RMWELHRSP/action/storage_attestation","attest_author":"https://pith.science/pith/TSMK42LLXYAMG6IN6RMWELHRSP/action/author_attestation","sign_citation":"https://pith.science/pith/TSMK42LLXYAMG6IN6RMWELHRSP/action/citation_signature","submit_replication":"https://pith.science/pith/TSMK42LLXYAMG6IN6RMWELHRSP/action/replication_record"}},"created_at":"2026-05-18T03:00:01.838728+00:00","updated_at":"2026-05-18T03:00:01.838728+00:00"}