{
  "id": "1610.04985",
  "version": "v1",
  "published": "2016-10-17T07:11:57.000Z",
  "updated": "2016-10-17T07:11:57.000Z",
  "title": "Some extensions of linear approximation and prediction problems for stationary processes",
  "authors": [
    "Ildar Ibragimov",
    "Zakhar Kabluchko",
    "Mikhail Lifshits"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(B(t))_{t\\in \\Theta}$ with $\\Theta={\\mathbb Z}$ or $\\Theta={\\mathbb R}$ be a wide sense stationary process with discrete or continuous time. The classical linear prediction problem consists of finding an element in $\\overline{span}\\{B(s),s\\le t\\}$ providing the best possible mean square approximation to the variable $B(\\tau)$ with $\\tau>t$. In this article we investigate this and some other similar problems where, in addition to prediction quality, optimization takes into account other features of the objects we search for. One of the most motivating examples of this kind is an approximation of a stationary process $B$ by a stationary differentiable process $X$ taking into account the kinetic energy that $X$ spends in its approximation efforts.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-17T07:11:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G10"
    ],
    "keywords": [
      "linear approximation",
      "extensions",
      "wide sense stationary process",
      "classical linear prediction problem consists",
      "mean square approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}