Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T10:39:10.009Z Has data issue: false hasContentIssue false
  • English
  • Français

The Gaussian distribution revisited

Part of: Classical measure theory Stochastic processes Limit theorems Turbulence

Published online by Cambridge University Press:  01 July 2016

Carlos E. Puente ,
Miguel M. López ,
Jorge E. Pinzón  and
José M. Angulo
Carlos E. Puente*
Affiliation:
University of California, Davis
Miguel M. López*
Affiliation:
University of British Columbia
Jorge E. Pinzón*
Affiliation:
University of California, Davis
José M. Angulo*
Affiliation:
Universidad de Granada
*
∗ Postal address: Hydrologic Science, Department of Land, Air and Water Resources, University of California, Davis, CA 95616, USA. Also at: Institute of Theoretical Dynamics, University of California, Davis, CA 95616, USA.
∗∗ Postal address: Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada.
∗∗∗ Postal address: Hydrologic Science, Department of Land, Air and Water Resources, University of California, Davis, CA 95616, USA.
∗∗∗∗ Postal address: Departamento de Estadística, Universidad de Granada, E-18071 Granada, Spain.
Get access Rights & Permissions [Opens in a new window]

Abstract

A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let fΘ,D:I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = fΘ,D(X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

Keywords

DETERMINISTIC FRACTAL GEOMETRYMULTIFRACTAL MEASURESFRACTAL INTERPOLATING FUNCTIONSGAUSSIAN DISTRIBUTION

MSC classification

Primary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 28A33: Spaces of measures, convergence of measures 28A80: Fractals
Secondary: 60F05: Central limit and other weak theorems 60G30: Continuity and singularity of induced measures 76F20: Dynamical systems approach to turbulence
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 500 - 524
Copyright
Copyright © Applied Probability Trust 1996 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barnsley, M. F. (1986) Fractal functions and interpolation. Constr. Approx. 2, 303–329.Google Scholar
Barnsley, M. F. (1988) Fractals Everywhere. Academic Press, New York.Google Scholar
Beck, C. and Roepstorff, G. (1987) From dynamical systems to the Langevin equation. Physica 145A, 1–14.Google Scholar
Billingsley, P. (1983) The singular function of bold play. Amer. Sci. 71, 392–397.Google Scholar
Billingsley, P. (1986) Probability and Measure. Wiley, New York.Google Scholar
Dubins, L. E. and Savage, L. J. (1960) Optimal gambling systems. PNAS 46, 1597–1598.Google Scholar
Feder, J. (1988) Fractals. Plenum, New York.CrossRefGoogle Scholar
Feller, W. (1968) An Introduction to Probability Theory and Its Applications. Vol. 1. 3rd edn. Wiley, London.Google Scholar
Kaye, B. H. (1989) A Random Walk Through Fractal Dimensions. Verlagsgesellschaft, Weinheim.Google Scholar
Mandelbrot, B. B. (1982) The Fractal Geometry of Nature. Freeman, San Fransicso.Google Scholar
Mandelbrot, B. B. (1989) Multifractal measures especially for the geophysicist. In Fractals in Geophysics. ed. Scholz, C. H. and Mandelbrot, B. B. Birkhauser, Basel.Google Scholar
Meneveau, C. and Sreenivasan, K. R. (1987) Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett. 59, 1424–1427.CrossRefGoogle ScholarPubMed
Phillip, W. and Stout, W. (1975) Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 161, 1–140.Google Scholar
Puente, C. E. (1992) Multinomial multifractals, fractal interpolators, and the Gaussian distribution. Phys. Lett. 161A, 441–447.Google Scholar
Puente, C. E. and Klebanoff, A. D. (1994) Gaussians everywhere. Fractals 2, 65–79.CrossRefGoogle Scholar
Ratner, M. (1973) The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Isr. J. Math. 16, 181–197.Google Scholar
Sreenivasan, K. R. (1991) Fractals and multifractals in fluid turbulence. Ann. Rev. Fluid Mech. 23, 539–600.Google Scholar