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We prove that $\\{(1-\\alpha) \\alpha^{\\gamma_\\alpha} {\\bf Z}_\\alpha^{-\\gamma_\\alpha}, 0< \\alpha <1\\}$ is decreasing for the optimal stochastic order and that $\\{(1-\\alpha){\\bf Z}_\\alpha^{-\\gamma_\\alpha}, 0< \\alpha < 1\\}$ is increasing for the convex order, with $\\gamma_\\alpha = \\alpha/(1-\\alpha).$ We also show that $\\{\\Gamma(1+\\alpha) {\\bf Z}_\\alpha^{-\\alpha}, 1/2\\le \\alpha \\le 1\\}$ is decreasing for the convex order, that ${\\bf Z}_\\alpha^{-\\alpha}\\,\\prec_{st}\\, \\Gamm"},"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":"1310.1888","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-07T19:01:53Z","cross_cats_sorted":[],"title_canon_sha256":"5f003c5d2fe22f639b457581305e974989a96f618d6677c863c22407ba727233","abstract_canon_sha256":"ab2e8fa0cad06f17bc2a78fc95b3e7417fbd2dfa5736e2e1f4e074b10b6e9178"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:01:10.130182Z","signature_b64":"Gi7H5Z9JJ+sstryOHbq2xwNhJmoXOLLiiwvZ+yHUqAf39FeoGJQeXE+ZQNfrKqC0eB2Vl04a/kHrctABgncNAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a240ebb7d2de6612dc9197042e5323ec22a808da5f27ebca3a8975608eaa8ac5","last_reissued_at":"2026-05-18T03:01:10.129445Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:01:10.129445Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Comparing Fr\\'echet and positive stable laws","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"LPTMS), Thomas Simon (LPP","submitted_at":"2013-10-07T19:01:53Z","abstract_excerpt":"Let ${\\bf L}$ be the unit exponential random variable and ${\\bf Z}_\\alpha$ the standard positive $\\alpha$-stable random variable. 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