{
  "id": "math/0612747",
  "version": "v2",
  "published": "2006-12-24T00:31:52.000Z",
  "updated": "2007-12-11T14:41:05.000Z",
  "title": "Law of the iterated logarithm for stationary processes",
  "authors": [
    "Ou Zhao",
    "Michael Woodroofe"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/009117907000000079 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2008, Vol. 36, No. 1, 127-142",
  "doi": "10.1214/009117907000000079",
  "categories": [
    "math.PR"
  ],
  "abstract": "There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes $...,X_{-1},X_0,X_1,...$ whose partial sums $S_n=X_1+...+X_n$ are of the form $S_n=M_n+R_n$, where $M_n$ is a square integrable martingale with stationary increments and $R_n$ is a remainder term for which $E(R_n^2)=o(n)$. Here we explore the law of the iterated logarithm (LIL) for the same class of processes. Letting $\\Vert\\cdot\\Vert$ denote the norm in $L^2(P)$, a sufficient condition for the partial sums of a stationary process to have the form $S_n=M_n+R_n$ is that $n^{-3/2}\\Vert E(S_n|X_0,X_{-1},...)\\Vert$ be summable. A sufficient condition for the LIL is only slightly stronger, requiring $n^{-3/2}\\log^{3/2}(n)\\Vert E(S_n|X_0,X_{-1},...)\\Vert$ to be summable. As a by-product of our main result, we obtain an improved statement of the conditional central limit theorem. Invariance principles are obtained as well.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-12-11T14:41:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "60F05"
    ],
    "keywords": [
      "iterated logarithm",
      "stationary processes",
      "sufficient condition",
      "conditional central limit theorem",
      "partial sums"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....12747Z"
    }
  }
}