{
  "id": "1407.6275",
  "version": "v2",
  "published": "2014-07-23T15:52:12.000Z",
  "updated": "2014-12-23T08:47:41.000Z",
  "title": "Mandelbrot cascades on random weighted trees and nonlinear smoothing transforms",
  "authors": [
    "Julien Barral",
    "Jacques Peyrière"
  ],
  "comment": "30 pages. This version contains a functional central limit theorem in the quadratic case",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider complex Mandelbrot multiplicative cascades on a random weigh\\-ted tree. Under suitable assumptions, this yields a dynamics $\\T$ on laws invariant by random weighted means (the so called fixed points of smoothing transformations) and which have a finite moment of order 2. Moreover, we can exhibit two main behaviors: If the weights are conservative, i.e., sum up to~1 almost surely, we find a domain for the initial law $\\mu$ such that a non-standard (functional) central limit theorem is valid for the orbit $(\\T^n\\mu)_{n\\ge 0}$ (this completes in a non trivial way our previous result in the case of non-negative Mandelbrot cascades on a regular tree). If the weights are non conservative, we find a domain for the initial law $\\mu$ over which $(\\T^n\\mu)_{n\\ge 0}$ converges to the law of a non trivial random variable whose law turns out to be a fixed point of a quadratic smoothing transformation, which naturally extends the usual notion of (linear) smoothing transformation; moreover, this limit law can be built as the limit of a non-negative martingale. Also, the dynamics can be modified to build fixed points of higher degree smoothing transformations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-23T15:52:12.000Z",
      "abstract": "We consider complex Mandelbrot multiplicative cascades on a random weigh\\-ted tree. Under suitable assumptions, this yields a dynamics $\\T$ on laws invariant by random weighted means (the so called fixed points of smoothing transformations) and which have a finite moment of order 2. Moreover, we can exhibit two main behaviors: If the weights are conservative, i.e. sum up to~1 almost surely, we find a domain for the initial law $\\mu$ such that a non-standard central limit theorem is valid for the orbit $(\\T^n\\mu)_{n\\ge 0}$ (this completes in a non trivial way our previous result in the case of non-negative Mandelbrot cascades on a regular tree). If the weights are non conservative, we find a domain for the initial law $\\mu$ over which $(\\T^n\\mu)_{n\\ge 0}$ converges to the law of a non trivial random variable whose law turns out to be a fixed point of a quadratic smoothing transformation, which naturally extends the usual notion of (linear) smoothing transformation; moreover, this limit law can be built as the limit of a non-negative martingale. Also, the dynamics can be modified to build fixed points of higher degrees smoothing transformations.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-23T08:47:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random weighted trees",
      "nonlinear smoothing transforms",
      "mandelbrot cascades",
      "smoothing transformation",
      "fixed point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.6275B"
    }
  }
}