{
  "id": "1601.01841",
  "version": "v1",
  "published": "2016-01-08T12:02:22.000Z",
  "updated": "2016-01-08T12:02:22.000Z",
  "title": "Expected number of real roots of random trigonometric polynomials",
  "authors": [
    "Hendrik Flasche"
  ],
  "comment": "15 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials $$ X_n(t)=u+\\frac{1}{\\sqrt{n}}\\sum_{k=1}^n (A_k\\cos(kt)+B_k\\sin(kt)), \\quad t\\in [0,2\\pi],\\quad u\\in\\mathbb{R} $$ whose coefficients $A_k, B_k$, $k\\in\\mathbb{N}$, are independent identically distributed random variables with zero mean and unit variance. If $N_n[a, b]$ denotes the number of real roots of $X_n$ in an interval $[a,b]\\subseteq [0,2\\pi]$, we prove that $$ \\lim_{n\\rightarrow\\infty} \\frac{\\mathbb{E} N_n[a,b]}{n}=\\frac{b-a}{\\pi\\sqrt{3}} e^{-\\frac{u^2}{2}}. $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-08T12:02:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26C10",
      "30C15",
      "42A05",
      "60F99",
      "60G15"
    ],
    "keywords": [
      "random trigonometric polynomials",
      "real roots",
      "expected number",
      "independent identically distributed random variables",
      "zero mean"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160101841F"
    }
  }
}