{
  "id": "1706.01654",
  "version": "v1",
  "published": "2017-06-06T08:33:08.000Z",
  "updated": "2017-06-06T08:33:08.000Z",
  "title": "On the real zeros of random trigonometric polynomials with dependent coefficients",
  "authors": [
    "Jürgen Angst",
    "Federico Dalmao",
    "Guillaume Poly"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "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 not only covers the well-known situation of independent coefficients but allow us to deal with long range correlations. In particular it englobes the case where the random coefficients are given by a fractional Brownian noise with any Hurst parameter.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-06T08:33:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26C10",
      "30C15",
      "42A05",
      "60F17",
      "60G55"
    ],
    "keywords": [
      "random trigonometric polynomials",
      "real zeros",
      "dependent coefficients",
      "independent stationary gaussian processes",
      "fractional brownian noise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}