{
  "id": "2001.03718",
  "version": "v1",
  "published": "2020-01-11T06:38:20.000Z",
  "updated": "2020-01-11T06:38:20.000Z",
  "title": "Fluctuations for matrix-valued Gaussian processes",
  "authors": [
    "Mario Diaz",
    "Arturo Jaramillo",
    "Juan Carlos Pardo"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a symmetric matrix-valued Gaussian process $Y^{(n)}=(Y^{(n)}(t);t\\ge0)$ and its empirical spectral measure process $\\mu^{(n)}=(\\mu_{t}^{(n)};t\\ge0)$. Under some mild conditions on the covariance function of $Y^{(n)}$, we find an explicit expression for the limit distribution of $$Z_F^{(n)} := \\left( \\big(Z_{f_1}^{(n)}(t),\\ldots,Z_{f_r}^{(n)}(t)\\big) ; t\\ge0\\right),$$ where $F=(f_1,\\dots, f_r)$, for $r\\ge 1$, with each component belonging to a large class of test functions, and $$ Z_{f}^{(n)}(t) := n\\int_{\\mathbb{R}}f(x)\\mu_{t}^{(n)}(\\text{d} x)-n\\mathbb{E}\\left[\\int_{\\mathbb{R}}f(x)\\mu_{t}^{(n)}(\\text{d} x)\\right].$$ More precisely, we establish the stable convergence of $Z_F^{(n)}$ and determine its limiting distribution. An upper bound for the total variation distance of the law of $Z_{f}^{(n)}(t)$ to its limiting distribution, for a test function $f$ and $t\\geq0$ fixed, is also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-11T06:38:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60B20",
      "60F05",
      "60H07",
      "60H05"
    ],
    "keywords": [
      "fluctuations",
      "test function",
      "symmetric matrix-valued gaussian process",
      "total variation distance",
      "limiting distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}