{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:2VDMOXZZ55JB3TLFVUGQWUOFLZ","short_pith_number":"pith:2VDMOXZZ","schema_version":"1.0","canonical_sha256":"d546c75f39ef521dcd65ad0d0b51c55e4b2d9bfd7df3c80957f07648c317e197","source":{"kind":"arxiv","id":"1607.06586","version":4},"attestation_state":"computed","paper":{"title":"Sample Variance in Free Probability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.OA","authors_text":"Franz Lehner, Wiktor Ejsmont","submitted_at":"2016-07-22T07:59:11Z","abstract_excerpt":"Let $X_1, X_2,\\dots, X_n$ denote i.i.d.~centered standard normal random variables, then the law of the sample variance $Q_n=\\sum_{i=1}^n(X_i-\\bar{X})^2$ is the $\\chi^2$-distribution with $n-1$ degrees of freedom. It is an open problem in classical probability to characterize all distributions with this property and in particular, whether it characterizes the normal law. In this paper we present a solution of the free analog of this question and show that the only distributions, whose free sample variance is distributed according to a free $\\chi^2$-distribution, are the semicircle law and more "},"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":"1607.06586","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2016-07-22T07:59:11Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"47e35a96f2e6e0f86ee75f5d5210652fb6c32032e3d88142b78adf563fb0f4bd","abstract_canon_sha256":"fd8c8e93956f3aa52ccdf97c28e503e3be6120860cd9a34cd1e742bc51668394"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:32.761788Z","signature_b64":"qASV4ql3sEEEvmc4Y0+FyZ4/2A2OP01FspVJo53O7vmXQXY5UHZQn9JqSa5KrwdDrsj0K4iAMpvW7T2ovgHyCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d546c75f39ef521dcd65ad0d0b51c55e4b2d9bfd7df3c80957f07648c317e197","last_reissued_at":"2026-05-17T23:39:32.761101Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:32.761101Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sample Variance in Free Probability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.OA","authors_text":"Franz Lehner, Wiktor Ejsmont","submitted_at":"2016-07-22T07:59:11Z","abstract_excerpt":"Let $X_1, X_2,\\dots, X_n$ denote i.i.d.~centered standard normal random variables, then the law of the sample variance $Q_n=\\sum_{i=1}^n(X_i-\\bar{X})^2$ is the $\\chi^2$-distribution with $n-1$ degrees of freedom. It is an open problem in classical probability to characterize all distributions with this property and in particular, whether it characterizes the normal law. In this paper we present a solution of the free analog of this question and show that the only distributions, whose free sample variance is distributed according to a free $\\chi^2$-distribution, are the semicircle law and more "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06586","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":"1607.06586","created_at":"2026-05-17T23:39:32.761204+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.06586v4","created_at":"2026-05-17T23:39:32.761204+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06586","created_at":"2026-05-17T23:39:32.761204+00:00"},{"alias_kind":"pith_short_12","alias_value":"2VDMOXZZ55JB","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"2VDMOXZZ55JB3TLF","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"2VDMOXZZ","created_at":"2026-05-18T12:29:55.572404+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/2VDMOXZZ55JB3TLFVUGQWUOFLZ","json":"https://pith.science/pith/2VDMOXZZ55JB3TLFVUGQWUOFLZ.json","graph_json":"https://pith.science/api/pith-number/2VDMOXZZ55JB3TLFVUGQWUOFLZ/graph.json","events_json":"https://pith.science/api/pith-number/2VDMOXZZ55JB3TLFVUGQWUOFLZ/events.json","paper":"https://pith.science/paper/2VDMOXZZ"},"agent_actions":{"view_html":"https://pith.science/pith/2VDMOXZZ55JB3TLFVUGQWUOFLZ","download_json":"https://pith.science/pith/2VDMOXZZ55JB3TLFVUGQWUOFLZ.json","view_paper":"https://pith.science/paper/2VDMOXZZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.06586&json=true","fetch_graph":"https://pith.science/api/pith-number/2VDMOXZZ55JB3TLFVUGQWUOFLZ/graph.json","fetch_events":"https://pith.science/api/pith-number/2VDMOXZZ55JB3TLFVUGQWUOFLZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2VDMOXZZ55JB3TLFVUGQWUOFLZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2VDMOXZZ55JB3TLFVUGQWUOFLZ/action/storage_attestation","attest_author":"https://pith.science/pith/2VDMOXZZ55JB3TLFVUGQWUOFLZ/action/author_attestation","sign_citation":"https://pith.science/pith/2VDMOXZZ55JB3TLFVUGQWUOFLZ/action/citation_signature","submit_replication":"https://pith.science/pith/2VDMOXZZ55JB3TLFVUGQWUOFLZ/action/replication_record"}},"created_at":"2026-05-17T23:39:32.761204+00:00","updated_at":"2026-05-17T23:39:32.761204+00:00"}