{
  "id": "0908.4472",
  "version": "v1",
  "published": "2009-08-31T08:07:28.000Z",
  "updated": "2009-08-31T08:07:28.000Z",
  "title": "On convergence to stationarity of fractional Brownian storage",
  "authors": [
    "Michel Mandjes",
    "Ilkka Norros",
    "Peter Glynn"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/08-AAP578 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 2009, Vol. 19, No. 4, 1385-1403",
  "doi": "10.1214/08-AAP578",
  "categories": [
    "math.PR"
  ],
  "abstract": "With $M(t):=\\sup_{s\\in[0,t]}A(s)-s$ denoting the running maximum of a fractional Brownian motion $A(\\cdot)$ with negative drift, this paper studies the rate of convergence of $\\mathbb {P}(M(t)>x)$ to $\\mathbb{P}(M>x)$. We define two metrics that measure the distance between the (complementary) distribution functions $\\mathbb{P}(M(t)>\\cdot)$ and $\\mathbb{P}(M>\\cdot)$. Our main result states that both metrics roughly decay as $\\exp(-\\vartheta t^{2-2H})$, where $\\vartheta$ is the decay rate corresponding to the tail distribution of the busy period in an fBm-driven queue, which was computed recently [Stochastic Process. Appl. (2006) 116 1269--1293]. The proofs extensively rely on application of the well-known large deviations theorem for Gaussian processes. We also show that the identified relation between the decay of the convergence metrics and busy-period asymptotics holds in other settings as well, most notably when G\\\"artner--Ellis-type conditions are fulfilled.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-08-31T08:07:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G18",
      "90B05"
    ],
    "keywords": [
      "fractional brownian storage",
      "convergence",
      "well-known large deviations theorem",
      "stationarity",
      "fractional brownian motion"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0908.4472M"
    }
  }
}