{
  "id": "1205.4172",
  "version": "v2",
  "published": "2012-05-18T15:07:26.000Z",
  "updated": "2013-10-21T06:52:58.000Z",
  "title": "Variance of partial sums of stationary sequences",
  "authors": [
    "George Deligiannidis",
    "Sergey Utev"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/12-AOP772 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2013, Vol. 41, No. 5, 3606-3616",
  "doi": "10.1214/12-AOP772",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X_1,X_2,\\ldots$ be a centred sequence of weakly stationary random variables with spectral measure $F$ and partial sums $S_n=X_1+\\cdots+X_n$. We show that $\\operatorname {var}(S_n)$ is regularly varying of index $\\gamma$ at infinity, if and only if $G(x):=\\int_{-x}^xF(\\mathrm {d}x)$ is regularly varying of index $2-\\gamma$ at the origin ($0<\\gamma<2$).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-21T06:52:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partial sums",
      "stationary sequences",
      "weakly stationary random variables",
      "spectral measure",
      "regularly varying"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.4172D"
    }
  }
}