{"paper":{"title":"Yule's \"Nonsense Correlation\" Solved!","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Abraham Wyner, Larry Shepp, Philip Ernst","submitted_at":"2016-08-14T17:46:08Z","abstract_excerpt":"In this paper, we resolve a longstanding open statistical problem. The problem is to mathematically confirm Yule's 1926 empirical finding of \"nonsense correlation\" (\\cite{Yule}). We do so by analytically determining the second moment of the empirical correlation coefficient\n  \\beqn \\theta := \\frac{\\int_0^1W_1(t)W_2(t) dt - \\int_0^1W_1(t) dt \\int_0^1 W_2(t) dt}{\\sqrt{\\int_0^1 W^2_1(t) dt - \\parens{\\int_0^1W_1(t) dt}^2} \\sqrt{\\int_0^1 W^2_2(t) dt - \\parens{\\int_0^1W_2(t) dt}^2}}, \\eeqn of two {\\em independent} Wiener processes, $W_1,W_2$. Using tools from Fred- holm integral equation theory, we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04120","kind":"arxiv","version":2},"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"}