{
  "id": "1801.07882",
  "version": "v1",
  "published": "2018-01-24T07:22:17.000Z",
  "updated": "2018-01-24T07:22:17.000Z",
  "title": "Invariance principle for non-homogeneous random walks",
  "authors": [
    "Nicholas Georgiou",
    "Aleksandar Mijatović",
    "Andrew R. Wade"
  ],
  "comment": "36 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in $\\mathbb{R}^d$, which may be recurrent in any dimension. The limit $\\mathcal{X}$ is an elliptic martingale diffusion, which may be point-recurrent at the origin for any $d\\geq2$. To characterise $\\mathcal{X}$, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in $\\mathbb{R}^d$ and use it to express a related spherical diffusion as a Brownian motion with drift. This representation allows us to establish the skew-product decomposition of the excursions of $\\mathcal{X}$ and thus develop the excursion theory of $\\mathcal{X}$ without appealing to the strong Markov property. This leads to the uniqueness in law of the stochastic differential equation for $\\mathcal{X}$ in $\\mathbb{R}^d$, whose coefficients are discontinuous at the origin. Using the Riemannian metric we can also detect whether the angular component of the excursions of $\\mathcal{X}$ is time-reversible. If so, the excursions of $\\mathcal{X}$ in $\\mathbb{R}^d$ generalise the classical Pitman-Yor splitting-at-the-maximum property of Bessel excursions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-24T07:22:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J05",
      "60J60",
      "60F17",
      "58J65",
      "60J55"
    ],
    "keywords": [
      "invariance principle",
      "riemannian metric",
      "classical pitman-yor splitting-at-the-maximum property",
      "elliptic martingale diffusion",
      "stochastic differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}