{
  "id": "0708.4183",
  "version": "v2",
  "published": "2007-08-30T15:17:44.000Z",
  "updated": "2008-11-14T07:16:47.000Z",
  "title": "On martingale approximations",
  "authors": [
    "Ou Zhao",
    "Michael Woodroofe"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/07-AAP505 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 2008, Vol. 18, No. 5, 1831-1847",
  "doi": "10.1214/07-AAP505",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider additive functionals of a Markov chain $W_k$, with stationary (marginal) distribution and transition function denoted by $\\pi$ and $Q$, say $S_n=g(W_1)+...+g(W_n)$, where $g$ is square integrable and has mean 0 with respect to $\\pi$. If $S_n$ has the form $S_n=M_n+R_n$, where $M_n$ is a square integrable martingale with stationary increments and $E(R_n^2)=o(n)$, then $g$ is said to admit a martingale approximation. Necessary and sufficient conditions for such an approximation are developed. Two obvious necessary conditions are $E[E(S_n|W_1)^2]=o(n)$ and $\\lim_{n\\to \\infty}E(S_n^2)/n<\\infty$. Assuming the first of these, let $\\Vert g\\Vert^2_+=\\limsup_{n\\to \\infty}E(S_n^2)/n$; then $\\Vert\\cdot\\Vert_+$ defines a pseudo norm on the subspace of $L^2(\\pi)$ where it is finite. In one main result, a simple necessary and sufficient condition for a martingale approximation is developed in terms of $\\Vert\\cdot\\Vert_+$. Let $Q^*$ denote the adjoint operator to $Q$, regarded as a linear operator from $L^2(\\pi)$ into itself, and consider co-isometries ($QQ^*=I$), an important special case that includes shift processes. In another main result a convenient orthonormal basis for $L_0^2(\\pi)$ is identified along with a simple necessary and sufficient condition for the existence of a martingale approximation in terms of the coefficients of the expansion of $g$ with respect to this basis.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-11-14T07:16:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60J10"
    ],
    "keywords": [
      "martingale approximation",
      "sufficient condition",
      "main result",
      "simple necessary",
      "important special case"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.4183Z"
    }
  }
}