{
  "id": "0707.3928",
  "version": "v2",
  "published": "2007-07-26T14:14:18.000Z",
  "updated": "2009-10-15T15:44:07.000Z",
  "title": "Asymptotic expansions for functions of the increments of certain Gaussian processes",
  "authors": [
    "Michael Marcus",
    "Jay Rosen"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $G=\\{G(x),x\\ge 0\\}$ be a mean zero Gaussian process with stationary increments and set $\\sigma^2(|x-y|)= E(G(x)-G(y))^2$. Let $f$ be a function with $Ef^{2}(\\eta)<\\ff$, where $\\eta=N(0,1)$. When $\\sigma^2$ is regularly varying at zero and \\[ \\lim_{h\\to 0}{h^2\\over \\sigma^2(h)}= 0\\qquad {and}\\qquad \\lim_{h\\to 0}{\\sigma^2(h)\\over h}= 0 \\quad {but} \\quad ({d^{2}\\over ds^2}\\sigma^2(s))^{j_0} \\] is locally integrable for some integer $j_0\\ge 1$, and satisfies some additional regularity conditions, \\bea && \\int_a^bf(\\frac{G(x+h)-G(x)}{\\sigma (h)}) dx \\label{abst}\\nn &&\\qquad = \\sum_{j=0}^{j_0} (h/\\sigma(h))^{j} {E(H_{j}(\\eta) f(\\eta))\\over\\sqrt {j!}} :(G')^{j}:(I_{[a,b]}) +o({h\\over\\sigma (h)})^{j_0}\\nn \\eea in $L^2$. Here $H_j$ is the $j$-th Hermite polynomial. Also $:(G')^{j}:(I_{[a,b]})$ is a $j $-th order Wick power Gaussian chaos constructed from the Gaussian field $ G'(g) $, with covariance \\[ E(G'(g)G'(\\wt g)) = \\int \\int \\rho (x-y)g(x)\\wt g(y) dx dy\\label{3.7bqs}, \\] where $ \\rho(s)={1/2}{d^{2}\\over ds^2}\\sigma^2(s)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-10-15T15:44:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "F25"
    ],
    "keywords": [
      "gaussian processes",
      "asymptotic expansions",
      "power gaussian chaos",
      "increments",
      "th order wick power gaussian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0707.3928M"
    }
  }
}