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Under the assumption that the elements of $M_n$ have 4 matching moments with those of GOE (resp. GUE), we show that the process $X_n(t)=\\sqrt{\\frac{\\beta n}{2}}\\sum_{i=1}^{\\lfloor nt\\rfloor}(|y_i|^2-\\frac1n)$ converges weakly to the Brownian bridge for any $\\mathbf{x}$ such that $||x||_\\infty\\rightarrow 0$ as $n\\rightarrow \\infty$, where $\\beta=1$ for the real case and $\\be"},"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":"1211.2507","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-12T03:45:24Z","cross_cats_sorted":[],"title_canon_sha256":"c6225f6a8df386a0fb8f43da277c80a549c492355a89ce5af44b9303c557372d","abstract_canon_sha256":"96eb6dd962728387c1684c7a2a2216bcf3b64234ddb1b6dedfd8dbd225ce4232"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:47.506474Z","signature_b64":"BRBQz13UhWsk4U8Et+FVRyDbD/FTQ5jItIhQVgdjxFqqA1q4Fw/C0Ay/Bn9ovXcrheignicmycsgFdPUVNZtDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f83bcfbb68e13dcb268333343a0383b91903643f4d38167211d4371a78b35e4a","last_reissued_at":"2026-05-18T03:08:47.505696Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:47.505696Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Universality for a global property of the eigenvectors of Wigner matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Guangming Pan, Wang Zhou, Zhigang Bao","submitted_at":"2012-11-12T03:45:24Z","abstract_excerpt":"Let $M_n$ be an $n\\times n$ real (resp. complex) Wigner matrix and $U_n\\Lambda_n U_n^*$ be its spectral decomposition. Set $(y_1,y_2...,y_n)^T=U_n^*x$, where $x=(x_1,x_2,...,$ $x_n)^T$ is a real (resp. complex) unit vector. Under the assumption that the elements of $M_n$ have 4 matching moments with those of GOE (resp. 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