{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:7C74FYE4WQHASFKPV4ZCWXSRLC","short_pith_number":"pith:7C74FYE4","canonical_record":{"source":{"id":"1001.4684","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-01-26T13:47:49Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"0b2b004e556a44b2b9a4d668af89645e07079cac2a870fe373868ccbc6f4f3e2","abstract_canon_sha256":"3f98c397f9e05a7e8d18405af3d97c8c4e2f06c64d11c62dae8d66362413e2c5"},"schema_version":"1.0"},"canonical_sha256":"f8bfc2e09cb40e09154faf322b5e5158aa299a28a0c3ded89f5e2f1a5b4d1df6","source":{"kind":"arxiv","id":"1001.4684","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.4684","created_at":"2026-05-18T03:25:59Z"},{"alias_kind":"arxiv_version","alias_value":"1001.4684v3","created_at":"2026-05-18T03:25:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.4684","created_at":"2026-05-18T03:25:59Z"},{"alias_kind":"pith_short_12","alias_value":"7C74FYE4WQHA","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"7C74FYE4WQHASFKP","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"7C74FYE4","created_at":"2026-05-18T12:26:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:7C74FYE4WQHASFKPV4ZCWXSRLC","target":"record","payload":{"canonical_record":{"source":{"id":"1001.4684","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-01-26T13:47:49Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"0b2b004e556a44b2b9a4d668af89645e07079cac2a870fe373868ccbc6f4f3e2","abstract_canon_sha256":"3f98c397f9e05a7e8d18405af3d97c8c4e2f06c64d11c62dae8d66362413e2c5"},"schema_version":"1.0"},"canonical_sha256":"f8bfc2e09cb40e09154faf322b5e5158aa299a28a0c3ded89f5e2f1a5b4d1df6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:59.859194Z","signature_b64":"tJgy0pjmmuz6IYfFCB7bGlnSFCmPZABSiDJlf4D495yqp4O5/x8BFwhfUDC/u5RvmO9reDOnZ/ZrrhiZcUl5AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8bfc2e09cb40e09154faf322b5e5158aa299a28a0c3ded89f5e2f1a5b4d1df6","last_reissued_at":"2026-05-18T03:25:59.858714Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:59.858714Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1001.4684","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:25:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KELpJyLKks0V4V+T2MRGLmENkBzRICqMDh826Ec7i9lE/b4jOy6iXo+cYx+ecsAXzLUlPycQzhkjR6nDv22/DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T18:33:16.582400Z"},"content_sha256":"172279b9fceb9788b269d99665cb0acce9e2691f5908a590f3403672a554ceef","schema_version":"1.0","event_id":"sha256:172279b9fceb9788b269d99665cb0acce9e2691f5908a590f3403672a554ceef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:7C74FYE4WQHASFKPV4ZCWXSRLC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Beta-Product Convolutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Enkelejd Hashorva","submitted_at":"2010-01-26T13:47:49Z","abstract_excerpt":"Let R be a positive random variable independent of S which is beta distributed. In this paper we are interested on the relation between the distribution function of R and that of RS. For this model we derive first some distributional properties, and then investigate the lower tail asymptotics of RS when R is regularly varying at 0, and vice-versa. Our first application concerns the asymptotic behaviour of the componentwise sample minima related to an elliptical distributions. Further, we derive the lower tails asymptotic of the aggregated risk for bivariate polar distributions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.4684","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:25:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CXh14Z5gjJ2RZyTVb51n/MEHb0muOcnzQsZlKBDH1IHMoOpTZW0GKYn1+LP1WHD4wYFElO1IbaA87/Tk48/wBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T18:33:16.583052Z"},"content_sha256":"5e1d2db1fa8385b23af912d9d29ce13bbae5c3593fcb3bb21ef1f4914cf54952","schema_version":"1.0","event_id":"sha256:5e1d2db1fa8385b23af912d9d29ce13bbae5c3593fcb3bb21ef1f4914cf54952"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7C74FYE4WQHASFKPV4ZCWXSRLC/bundle.json","state_url":"https://pith.science/pith/7C74FYE4WQHASFKPV4ZCWXSRLC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7C74FYE4WQHASFKPV4ZCWXSRLC/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-07T18:33:16Z","links":{"resolver":"https://pith.science/pith/7C74FYE4WQHASFKPV4ZCWXSRLC","bundle":"https://pith.science/pith/7C74FYE4WQHASFKPV4ZCWXSRLC/bundle.json","state":"https://pith.science/pith/7C74FYE4WQHASFKPV4ZCWXSRLC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7C74FYE4WQHASFKPV4ZCWXSRLC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:7C74FYE4WQHASFKPV4ZCWXSRLC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3f98c397f9e05a7e8d18405af3d97c8c4e2f06c64d11c62dae8d66362413e2c5","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-01-26T13:47:49Z","title_canon_sha256":"0b2b004e556a44b2b9a4d668af89645e07079cac2a870fe373868ccbc6f4f3e2"},"schema_version":"1.0","source":{"id":"1001.4684","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.4684","created_at":"2026-05-18T03:25:59Z"},{"alias_kind":"arxiv_version","alias_value":"1001.4684v3","created_at":"2026-05-18T03:25:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.4684","created_at":"2026-05-18T03:25:59Z"},{"alias_kind":"pith_short_12","alias_value":"7C74FYE4WQHA","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"7C74FYE4WQHASFKP","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"7C74FYE4","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:5e1d2db1fa8385b23af912d9d29ce13bbae5c3593fcb3bb21ef1f4914cf54952","target":"graph","created_at":"2026-05-18T03:25:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let R be a positive random variable independent of S which is beta distributed. In this paper we are interested on the relation between the distribution function of R and that of RS. For this model we derive first some distributional properties, and then investigate the lower tail asymptotics of RS when R is regularly varying at 0, and vice-versa. Our first application concerns the asymptotic behaviour of the componentwise sample minima related to an elliptical distributions. Further, we derive the lower tails asymptotic of the aggregated risk for bivariate polar distributions.","authors_text":"Enkelejd Hashorva","cross_cats":["math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-01-26T13:47:49Z","title":"On Beta-Product Convolutions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.4684","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:172279b9fceb9788b269d99665cb0acce9e2691f5908a590f3403672a554ceef","target":"record","created_at":"2026-05-18T03:25:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3f98c397f9e05a7e8d18405af3d97c8c4e2f06c64d11c62dae8d66362413e2c5","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-01-26T13:47:49Z","title_canon_sha256":"0b2b004e556a44b2b9a4d668af89645e07079cac2a870fe373868ccbc6f4f3e2"},"schema_version":"1.0","source":{"id":"1001.4684","kind":"arxiv","version":3}},"canonical_sha256":"f8bfc2e09cb40e09154faf322b5e5158aa299a28a0c3ded89f5e2f1a5b4d1df6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8bfc2e09cb40e09154faf322b5e5158aa299a28a0c3ded89f5e2f1a5b4d1df6","first_computed_at":"2026-05-18T03:25:59.858714Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:25:59.858714Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tJgy0pjmmuz6IYfFCB7bGlnSFCmPZABSiDJlf4D495yqp4O5/x8BFwhfUDC/u5RvmO9reDOnZ/ZrrhiZcUl5AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:25:59.859194Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.4684","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:172279b9fceb9788b269d99665cb0acce9e2691f5908a590f3403672a554ceef","sha256:5e1d2db1fa8385b23af912d9d29ce13bbae5c3593fcb3bb21ef1f4914cf54952"],"state_sha256":"2fa28eb3068cc26cdc73687bbe0fcd08ee4ae05527ef39334b346e785b1d472f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2SrT7hk+eoR1BG17A4EtKGBiWMoSHZHZxaDY7TRipMiFS/SXE9hIaEFE4+XqgRIgFSkbIT9ia31cN+08TqeGDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T18:33:16.586881Z","bundle_sha256":"ff8e5416cd806c5b5446074da507d573a7517110e32ae064a0dbb64baf8d7472"}}