{
  "id": "1709.03753",
  "version": "v1",
  "published": "2017-09-12T09:18:58.000Z",
  "updated": "2017-09-12T09:18:58.000Z",
  "title": "AR(1) sequence with random coefficients: Regenerative properties and its application",
  "authors": [
    "Krishna B. Athreya",
    "Koushik Saha",
    "Radhendushka Srivastava"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{X_n\\}_{n\\ge0}$ be a sequence of real valued random variables such that $X_n=\\rho_n X_{n-1}+\\epsilon_n,~n=1,2,\\ldots$, where $\\{(\\rho_n,\\epsilon_n)\\}_{n\\ge1}$ are i.i.d. and independent of initial value (possibly random) $X_0$. In this paper it is shown that, under some natural conditions on the distribution of $(\\rho_1,\\epsilon_1)$, the sequence $\\{X_n\\}_{n\\ge0}$ is regenerative in the sense that it could be broken up into i.i.d. components. Further, when $\\rho_1$ and $\\epsilon_1$ are independent, we construct a non-parametric strongly consistent estimator of the characteristic functions of $\\rho_1$ and $\\epsilon_1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-12T09:18:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G20"
    ],
    "keywords": [
      "random coefficients",
      "regenerative properties",
      "application",
      "real valued random variables",
      "non-parametric strongly consistent estimator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}