{
  "id": "1711.03316",
  "version": "v1",
  "published": "2017-11-09T10:43:34.000Z",
  "updated": "2017-11-09T10:43:34.000Z",
  "title": "Non universality for the variance of the number of real roots of random trigonometric polynomials",
  "authors": [
    "Vlad Bally",
    "Lucia Caramellino",
    "Guillaume Poly"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article, we consider the following family of random trigonometric polynomials $p_n(t,Y)=\\sum_{k=1}^n Y_{k,1} \\cos(kt)+Y_{k,2}\\sin(kt)$ for a given sequence of i.i.d. random variables $\\{Y_{k,1},Y_{k,2}\\}_{k\\ge 1}$ which are centered and standardized. We set $\\mathcal{N}([0,\\pi],Y)$ the number of real roots over $[0,\\pi]$ and $\\mathcal{N}([0,\\pi],G)$ the corresponding quantity when the coefficients follow a standard Gaussian distribution. We prove under a Doeblin's condition on the distribution of the coefficients that $$ \\lim_{n\\to\\infty}\\frac{\\text{Var}\\left(\\mathcal{N}_n([0,\\pi],Y)\\right)}{n} =\\lim_{n\\to\\infty}\\frac{\\text{Var}\\left(\\mathcal{N}_n([0,\\pi],G)\\right)}{n} +\\frac{1}{30}\\left(\\mathbb{E}(Y_{1,1}^4)-3\\right). $$ The latter establishes that the behavior of the variance is not universal and depends on the distribution of the underlying coefficients through their kurtosis. Actually, a more general result is proven in this article, which does not require that the coefficients are identically distributed. The proof mixes a recent result regarding Edgeworth's expansions for distribution norms established in arXiv:1606.01629 with the celebrated Kac-Rice formula.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-09T10:43:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random trigonometric polynomials",
      "real roots",
      "non universality",
      "coefficients",
      "standard gaussian distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}