{
  "id": "1311.6923",
  "version": "v2",
  "published": "2013-11-27T10:38:41.000Z",
  "updated": "2015-10-09T09:32:22.000Z",
  "title": "Asymptotics of random processes with immigration II: convergence to stationarity",
  "authors": [
    "Alexander Iksanov",
    "Alexander Marynych",
    "Matthias Meiners"
  ],
  "comment": "20 pages, accepted for publication in Bernoulli",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X_1, X_2,\\ldots$ be random elements of the Skorokhod space $D(\\mathbb{R})$ and $\\xi_1, \\xi_2, \\ldots$ positive random variables such that the pairs $(X_1,\\xi_1), (X_2,\\xi_2),\\ldots$ are independent and identically distributed. We call the random process $(Y(t))_{t \\in \\mathbb{R}}$ defined by $Y(t):=\\sum_{k \\geq 0}X_{k+1}(t-\\xi_1-\\ldots-\\xi_k)1_{\\{\\xi_1+\\ldots+\\xi_k\\leq t\\}}$, $t\\in\\mathbb{R}$ random process with immigration at the epochs of a renewal process. Assuming that $X_k$ and $\\xi_k$ are independent and that the distribution of $\\xi_1$ is nonlattice and has finite mean we investigate weak convergence of $(Y(t))_{t\\in\\mathbb{R}}$ as $t\\to\\infty$ in $D(\\mathbb{R})$ endowed with the $J_1$-topology. The limits are stationary processes with immigration.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-27T10:38:41.000Z",
      "title": "Limit theorems for random processes with immigration at the epochs of a renewal process I",
      "abstract": "Let $X_1, X_2,\\ldots$ be random elements of the Skorokhod space $D[0,\\infty)$ and $\\xi_1, \\xi_2, \\ldots$ positive random variables such that the pairs $(X_1,\\xi_1), (X_2,\\xi_2),\\ldots$ are independent and identically distributed. We call the random process $(Y(t))_{t \\in \\mathbb{R}}$ defined by $Y(t):=\\sum_{k \\geq 0}X_{k+1}(t-\\xi_1-\\ldots-\\xi_k)1_{\\{\\xi_1+\\ldots+\\xi_k\\leq t\\}}$, $t\\geq 0$ random process with immigration at the epochs of a renewal process. Assuming that $X_k$ and $\\xi_k$ are independent, we investigate weak convergence of $(Y(t))_{t \\in \\mathbb{R}}$ when neither scaling, nor centering is needed. In the case when the distribution of $\\xi_1$ is nonlattice and has finite mean, we provide sufficient conditions for the weak convergence of the finite-dimensional distributions of $(Y(u+t))_{u\\in\\mathbb{R}}$ as $t\\to\\infty$ and sufficient conditions for the weak convergence of the same processes in the Skorokhod space endowed with the $J_1$-topology. The limits are stationary processes with immigration. In the case when the functions $\\mathbb{P}\\{\\xi_1>t\\}$ and ${\\rm Var}\\, [X_1(t)]$ are regularly varying at infinity of negative index strictly larger than $-1$ and asymptotically equivalent we derive sufficient conditions for the weak convergence of the finite-dimensional distributions of $(Y(ut))_{u>0}$ as $t\\to\\infty$. The limits are conditionally Gaussian processes, and any version of these processes takes values in the Skorokhod space $D(0,\\infty)$ with probability less than one.",
      "comment": "24 pages, submitted",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-10-09T09:32:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "renewal process",
      "random processes",
      "limit theorems",
      "weak convergence",
      "skorokhod space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.6923I"
    }
  }
}