{
  "id": "1405.0671",
  "version": "v2",
  "published": "2014-05-04T09:31:05.000Z",
  "updated": "2015-10-09T09:26:56.000Z",
  "title": "Asymptotics of random processes with immigration I: scaling limits",
  "authors": [
    "Alexander Iksanov",
    "Alexander Marynych",
    "Matthias Meiners"
  ],
  "comment": "46 pages, accepted for publication in Bernoulli",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(X_1, \\xi_1), (X_2,\\xi_2),\\ldots$ be i.i.d.~copies of a pair $(X,\\xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\\infty)$ and $\\xi$ is a positive random variable. Define $S_k := \\xi_1+\\ldots+\\xi_k$, $k \\in \\mathbb{N}_0$ and $Y(t) := \\sum_{k\\geq 0} X_{k+1}(t-S_k) 1_{\\{S_k \\leq t\\}}$, $t\\geq 0$. We call the process $(Y(t))_{t \\geq 0}$ random process with immigration at the epochs of a renewal process. We investigate weak convergence of the finite-dimensional distributions of $(Y(ut))_{u>0}$ as $t\\to\\infty$. Under the assumptions that the covariance function of $X$ is regularly varying in $(0,\\infty)\\times (0,\\infty)$ in a uniform way, the class of limiting processes is rather rich and includes Gaussian processes with explicitly given covariance functions, fractionally integrated stable L\\'evy motions and their sums when the law of $\\xi$ belongs to the domain of attraction of a stable law with finite mean, and conditionally Gaussian processes with explicitly given (conditional) covariance functions, fractionally integrated inverse stable subordinators and their sums when the law of $\\xi$ belongs to the domain of attraction of a stable law with infinite mean.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-04T09:31:05.000Z",
      "title": "Limit theorems for random processes with immigration at the epochs of a renewal process II",
      "abstract": "Let $(X_1, \\xi_1), (X_2,\\xi_2),\\ldots$ be i.i.d. copies of a tuple $(X,\\xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\\infty)$ and $\\xi$ is a positive random variable. Define $S_k := \\xi_1+\\ldots+\\xi_k$, $k \\in \\mathbb{N}_0$ and $Y(t) := \\sum_{k\\geq 0} X_{k+1}(t-S_k) 1_{\\{S_k \\leq t\\}}$, $t\\geq 0$. We call the process $(Y(t))_{t \\geq 0}$ random process with immigration at the epochs of a renewal process. We investigate weak convergence of finite-dimensional distributions of $(Y(ut))_{t \\geq 0}$, as $t \\to \\infty$ in the situations when a non-trivial normalization is needed. Under the assumption that the covariance function of $X$ is regularly varying in $(0,\\infty)\\times (0,\\infty)$ in a uniform way, the class of limiting processes is rather rich and includes Gaussian processes with explicitly given covariance functions, fractionally integrated stable L\\'evy motions and their sums when the law of $\\xi$ belongs to the domain of attraction of a stable law with finite mean, and conditionally Gaussian processes with explicitly given (conditional) covariance functions, fractionally integrated inverse stable subordinators and their sums when the law of $\\xi$ belongs to the domain of attraction of a stable law with infinite mean.",
      "comment": "31 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-10-09T09:26:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "renewal process",
      "random processes",
      "limit theorems",
      "integrated inverse stable subordinators",
      "integrated stable levy motions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.0671I"
    }
  }
}