{
  "id": "2107.00760",
  "version": "v1",
  "published": "2021-07-01T22:10:55.000Z",
  "updated": "2021-07-01T22:10:55.000Z",
  "title": "Functional limit theorems for random walks perturbed by positive alpha-stable jumps",
  "authors": [
    "Alexander Iksanov",
    "Andrey Pilipenko",
    "Oleksandr Prykhodko"
  ],
  "comment": "submitted for publication, 22 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\xi_1$, $\\xi_2,\\ldots$ be i.i.d. random variables of zero mean and finite variance and $\\eta_1$, $\\eta_2,\\ldots$ positive i.i.d. random variables whose distribution belongs to the domain of attraction of an $\\alpha$-stable distribution, $\\alpha\\in (0,1)$. The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump $\\xi_k$ occurs; if the present position of the Markov chain is nonpositive, then its next position is $\\eta_j$. We prove a functional limit theorem for this Markov chain under Donsker's scaling. The weak limit is a nonnegative process $(X(t))_{t\\geq 0}$ satisfying a stochastic equation ${\\rm d}X(t)={\\rm d}W(t)+ {\\rm d}U_\\alpha(L_X^{(0)}(t))$, where $W$ is a Brownian motion, $U_\\alpha$ is an $\\alpha$-stable subordinator which is independent of $W$, and $L_X^{(0)}$ is a local time of $X$ at $0$. Also, we explain that $X$ is a Feller Brownian motion with a `jump-type' exit from $0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-01T22:10:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "functional limit theorem",
      "positive alpha-stable jumps",
      "markov chain",
      "random walks",
      "random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}