{
  "id": "0903.2738",
  "version": "v2",
  "published": "2009-03-16T13:39:17.000Z",
  "updated": "2010-11-15T09:45:43.000Z",
  "title": "Stationary systems of Gaussian processes",
  "authors": [
    "Zakhar Kabluchko"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/10-AAP686 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2010, Vol. 20, No. 6, 2295-2317",
  "doi": "10.1214/10-AAP686",
  "categories": [
    "math.PR"
  ],
  "abstract": "We describe all countable particle systems on $\\mathbb{R}$ which have the following three properties: independence, Gaussianity and stationarity. More precisely, we consider particles on the real line starting at the points of a Poisson point process with intensity measure $\\mathfrak{m}$ and moving independently of each other according to the law of some Gaussian process $\\xi$. We classify all pairs $(\\mathfrak{m},\\xi)$ generating a stationary particle system, obtaining three families of examples. In the first, trivial family, the measure $\\mathfrak{m}$ is arbitrary, whereas the process $\\xi$ is stationary. In the second family, the measure $\\mathfrak{m}$ is a multiple of the Lebesgue measure, and $\\xi$ is essentially a Gaussian stationary increment process with linear drift. In the third, most interesting family, the measure $\\mathfrak{m}$ has a density of the form $\\alpha e^{-\\lambda x}$, where $\\alpha >0$, $\\lambda\\in\\mathbb{R}$, whereas the process $\\xi$ is of the form $\\xi(t)=W(t)-\\lambda\\sigma ^2(t)/2+c$, where $W$ is a zero-mean Gaussian process with stationary increments, $\\sigma ^2(t)=\\operatorname {Var}W(t)$, and $c\\in\\mathbb{R}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-11-15T09:45:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian processes",
      "stationary systems",
      "gaussian stationary increment process",
      "stationary particle system",
      "zero-mean gaussian process"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.2738K"
    }
  }
}