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We show that if the linear statistics $\\frac{1}{a_n} \\left[ f\\left( \\frac {Z^{(n)}_1}{b_n} \\right)+ \\cdots + f \\left(\\frac {Z^{(n)}_n}{b_n} \\right) \\right]$ associated with $\\{Z^{(n)}_j\\}$ has a limit as $n\\to\\infty$ at some mode of convergence, the linear statistics associated with $\\{X^{n, k}_j\\}$ converges to the same limit at the same mode. Similar statement also holds for the centered line"},"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":"1701.03946","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-01-14T16:44:02Z","cross_cats_sorted":[],"title_canon_sha256":"c4213e53a7d2d8ec43114cb708665e79f985c132845f82ce3a36fb6b20c87520","abstract_canon_sha256":"b86b0c51367c308d3aeed01e5ac348a2da1c9ad86d8af5a5eb345d62f4fda980"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:48.332120Z","signature_b64":"OeNAqiL+62ieqELnPXINMQJu/lzOBhruNmkYhBDMu5Ib2wY1BOyxyhnfqCjYBUrim+S58wf0Mi5CkhMQHSQrBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63cb515d7118d606cfe54485d0609e1858af978d3f22261adc0e170705ef8cc3","last_reissued_at":"2026-05-18T00:52:48.331566Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:48.331566Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Common Limit of the Linear Statistics of Zeros of Random Polynomials and Their Derivatives","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chih-Chung Chang, I-Shing Hu","submitted_at":"2017-01-14T16:44:02Z","abstract_excerpt":"Let $ p_n(x) $ be a random polynomial of degree $n$ and $\\{Z^{(n)}_j\\}_{j=1}^n$ and $\\{X^{n, k}_j\\}_{j=1}^{n-k}, k<n$, be the zeros of $p_n$ and $p_n^{(k)}$, the $k$th derivative of $p_n$, respectively. We show that if the linear statistics $\\frac{1}{a_n} \\left[ f\\left( \\frac {Z^{(n)}_1}{b_n} \\right)+ \\cdots + f \\left(\\frac {Z^{(n)}_n}{b_n} \\right) \\right]$ associated with $\\{Z^{(n)}_j\\}$ has a limit as $n\\to\\infty$ at some mode of convergence, the linear statistics associated with $\\{X^{n, k}_j\\}$ converges to the same limit at the same mode. Similar statement also holds for the centered line"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.03946","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":"1701.03946","created_at":"2026-05-18T00:52:48.331647+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.03946v1","created_at":"2026-05-18T00:52:48.331647+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.03946","created_at":"2026-05-18T00:52:48.331647+00:00"},{"alias_kind":"pith_short_12","alias_value":"MPFVCXLRDDLA","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_16","alias_value":"MPFVCXLRDDLANT7F","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_8","alias_value":"MPFVCXLR","created_at":"2026-05-18T12:31:31.346846+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/MPFVCXLRDDLANT7FISC5AYE6DB","json":"https://pith.science/pith/MPFVCXLRDDLANT7FISC5AYE6DB.json","graph_json":"https://pith.science/api/pith-number/MPFVCXLRDDLANT7FISC5AYE6DB/graph.json","events_json":"https://pith.science/api/pith-number/MPFVCXLRDDLANT7FISC5AYE6DB/events.json","paper":"https://pith.science/paper/MPFVCXLR"},"agent_actions":{"view_html":"https://pith.science/pith/MPFVCXLRDDLANT7FISC5AYE6DB","download_json":"https://pith.science/pith/MPFVCXLRDDLANT7FISC5AYE6DB.json","view_paper":"https://pith.science/paper/MPFVCXLR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.03946&json=true","fetch_graph":"https://pith.science/api/pith-number/MPFVCXLRDDLANT7FISC5AYE6DB/graph.json","fetch_events":"https://pith.science/api/pith-number/MPFVCXLRDDLANT7FISC5AYE6DB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MPFVCXLRDDLANT7FISC5AYE6DB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MPFVCXLRDDLANT7FISC5AYE6DB/action/storage_attestation","attest_author":"https://pith.science/pith/MPFVCXLRDDLANT7FISC5AYE6DB/action/author_attestation","sign_citation":"https://pith.science/pith/MPFVCXLRDDLANT7FISC5AYE6DB/action/citation_signature","submit_replication":"https://pith.science/pith/MPFVCXLRDDLANT7FISC5AYE6DB/action/replication_record"}},"created_at":"2026-05-18T00:52:48.331647+00:00","updated_at":"2026-05-18T00:52:48.331647+00:00"}