{
  "id": "1201.4579",
  "version": "v1",
  "published": "2012-01-22T17:31:11.000Z",
  "updated": "2012-01-22T17:31:11.000Z",
  "title": "Limit theorems for stationary Markov processes with L2-spectral gap",
  "authors": [
    "Deborah Ferre",
    "Loïc Hervé",
    "James Ledoux"
  ],
  "comment": "35 pages Accepted(6 january 2011) for publication in Annales de l'Institut Henri Poincare - Probabilites et Statistiques",
  "journal": "Annales de l'IHP - Probabilit\\'es et Statistiques 48, 2 (2012) 396-423",
  "doi": "10.1214/11-AIHP413",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Let $(X_t, Y_t)_{t\\in T}$ be a discrete or continuous-time Markov process with state space $X \\times R^d$ where $X$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. $(X_t, Y_t)_{t\\in T}$ is assumed to be a Markov additive process. In particular, this implies that the first component $(X_t)_{t\\in T}$ is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process $(Y_t)_{t\\in T}$ is shown to satisfy the following classical limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c) the one-dimensional Berry-Esseen theorem, (d) the one-dimensional first-order Edgeworth expansion, provided that we have sup{t\\in(0,1]\\cap T : E{\\pi,0}[|Y_t| ^{\\alpha}] < 1 with the expected order with respect to the independent case (up to some $\\varepsilon > 0$ for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process $(X_t)_{t\\in T}$ has an invariant probability distribution $\\pi$, is stationary and has the $L^2(\\pi)$-spectral gap property (that is, $(X_t)t\\in N}$ is $\\rho$-mixing in the discrete-time case). The case where $(X_t)_{t\\in T}$ is non-stationary is briefly discussed. As an application, we derive a Berry-Esseen bound for the M-estimators associated with $\\rho$-mixing Markov chains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-22T17:31:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stationary markov processes",
      "l2-spectral gap",
      "independent case",
      "one-dimensional first-order edgeworth expansion",
      "spectral gap property"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012AnIHP..48..396F"
    }
  }
}