{"paper":{"title":"On the real zeros of random trigonometric polynomials with dependent coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Federico Dalmao, Guillaume Poly, J\\\"urgen Angst","submitted_at":"2017-06-06T08:33:08Z","abstract_excerpt":"We consider random trigonometric polynomials of the form \\[ f_n(t):=\\sum_{1\\le k \\le n} a_{k} \\cos(kt) + b_{k} \\sin(kt), \\] whose entries $(a_{k})_{k\\ge 1}$ and $(b_{k})_{k\\ge 1}$ are given by two independent stationary Gaussian processes with the same correlation function $\\rho$. Under mild assumptions on the spectral function $\\psi_\\rho$ associated with $\\rho$, we prove that the expectation of the number $N_n([0,2\\pi])$ of real roots of $f_n$ in the interval $[0,2\\pi]$ satisfies \\[ \\lim_{n \\to +\\infty} \\frac{\\mathbb E\\left [N_n([0,2\\pi])\\right]}{n} = \\frac{2}{\\sqrt{3}}. \\] The latter result "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.01654","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"}