{
  "id": "1810.09861",
  "version": "v1",
  "published": "2018-10-23T13:52:11.000Z",
  "updated": "2018-10-23T13:52:11.000Z",
  "title": "Persistence exponents via perturbation theory: AR(1)-processes",
  "authors": [
    "Frank Aurzada",
    "Marvin Kettner"
  ],
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "For AR(1)-processes $X_n=\\rho X_{n-1}+\\xi_n$, $n\\in\\mathbb{N}$, where $\\rho\\in\\mathbb{R}$ and $(\\xi_i)_{i\\in\\mathbb{N}}$ is an i.i.d. sequence of random variables, we study the persistence probabilities $\\mathbb{P}(X_0\\ge 0,\\dots, X_N\\ge 0)$ for $N\\to\\infty$. For a wide class of Markov processes a recent result [Aurzada, Mukherjee, Zeitouni; arXiv:1703.06447; 2017] shows that these probabilities decrease exponentially fast and that the rate of decay can be identified as an eigenvalue of some integral operator. We discuss a perturbation technique to determine a series expansion of the eigenvalue in the parameter $\\rho$ for normally distributed AR(1)-processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-23T13:52:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "perturbation theory",
      "persistence exponents",
      "probabilities decrease exponentially fast",
      "series expansion",
      "markov processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}