{
  "id": "1610.09965",
  "version": "v1",
  "published": "2016-10-31T15:23:58.000Z",
  "updated": "2016-10-31T15:23:58.000Z",
  "title": "Stability of perpetuities in Markovian environment",
  "authors": [
    "Gerold Alsmeyer",
    "Fabian Buckmann"
  ],
  "comment": "36 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "The stability of iterations of affine linear maps $\\Psi_{n}(x)=A_{n}x+B_{n}$, $n=1,2,\\ldots$, is studied in the presence of a Markovian environment, more precisely, for the situation when $(A_{n},B_{n})_{n\\ge 1}$ is modulated by an ergodic Markov chain $(M_{n})_{n\\ge 0}$ with countable state space $\\mathcal{S}$ and stationary distribution $\\pi$. We provide necessary and sufficient conditions for the a.s. and the distributional convergence of the backward iterations $\\Psi_{1}\\circ\\ldots\\circ\\Psi_{n}(Z_{0})$ and also describe all possible limit laws as solutions to a certain Markovian stochastic fixed-point equation. As a consequence of the random environment, these limit laws are stochastic kernels from $\\mathcal{S}$ to $\\mathbb{R}$ rather than distributions on $\\mathbb{R}$, thus reflecting their dependence on where the driving chain is started. We give also necessary and sufficient conditions for the distributional convergence of the forward iterations $\\Psi_{n}\\circ\\ldots\\circ\\Psi_{1}$. The main differences caused by the Markovian environment as opposed to the extensively studied case of independent and identically distributed (iid) $\\Psi_{1},\\Psi_{2},\\ldots$ are that: (1) backward iterations may still converge in distribution, if a.s. convergence fails, (2) the degenerate case when $A_{1}c_{M_{1}}+B_{1}=c_{M_{0}}$ a.s. for suitable constants $c_{i}$, $i\\in\\mathcal{S}$, is by far more complex than the degenerate case for iid $(A_{n},B_{n})$ when $A_{1}c+B_{1}=c$ a.s. for some $c\\in\\mathbb{R}$, and (3) forward and backward iterations generally have different laws given $M_{0}=i$ for $i\\in\\mathcal{S}$ so that the former ones need a separate analysis. Our proofs draw on related results for the iid-case, notably by Vervaat, Grincevi\\v{c}ius, and Goldie and Maller, in combination with recent results by the authors on fluctuation theory for Markov random walks.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-31T15:23:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60H25",
      "60J15",
      "60K05",
      "60K15"
    ],
    "keywords": [
      "markovian environment",
      "backward iterations",
      "perpetuities",
      "markovian stochastic fixed-point equation",
      "limit laws"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}