• Bernoulli
  • Volume 5, Number 2 (1999), 191-207.

Estimation of Rényi exponents in random cascades

Brent M. Troutman and Aldo V. Vecchia

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We consider statistical estimation of the Rényi exponent τ (h) , which characterizes the scaling behaviour of a singular measure μ defined on a subset of R d . The Rényi exponent is defined to be lim δ 0 [{logM δ (h)}/(-logδ)] , assuming that this limit exists, where M δ (h)= i μ h(Δ i) and, for δ >0 , { Δ i} are the cubes of a δ -coordinate mesh that intersect the support of μ . In particular, we demonstrate asymptotic normality of the least-squares estimator of τ (h) when the measure μ is generated by a particular class of multiplicative random cascades, a result which allows construction of interval estimates and application of hypothesis tests for this scaling exponent. Simulation results illustrating this asymptotic normality are presented.

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Bernoulli, Volume 5, Number 2 (1999), 191-207.

First available in Project Euclid: 5 March 2007

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least-squares estimation multifractal multiplicative process random cascade Rényi exponent scaling exponent


Troutman, Brent M.; Vecchia, Aldo V. Estimation of Rényi exponents in random cascades. Bernoulli 5 (1999), no. 2, 191--207.

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  • [1] Cates, M.E. and Deutsch, J.M. (1987) Spatial correlations in multifractals. Phys. Rev. A, 35, 4907-4910.
  • [2] Davis, A., Marshak, A. and Wiscombe, W.J. (1993) Bi-multifractal analysis and multi-affine modeling of non-stationary geophysical processes, application to turbulence and clouds. Fractals, 1, 560-567.
  • [3] Gupta, V.K. and Waymire, E.C. (1990) Multiscaling properties of spatial rainfall and river flow distributions. J. Geophys. Res., 95, 1999-2009.
  • [4] Gupta, V.K. and Waymire, E.C. (1993) A statistical analysis of mesoscale rainfall as a random cascade. J. Appl. Meteorol., 32, 251-267.
  • [5] Gupta, V.K. and Waymire, E.C. (1996) Multiplicative cascades and spatial variability in rainfall, river networks, and floods. In J.B. Rundle, D.L. Turcotte and W. Klein (eds), Reduction and Predictability of Natural Disasters, pp. 71-96. Santa Fe Institute Proc., Vol. XXV. Reading, MA: Addison-Wesley.
  • [6] Halsey, T.C., Jensen, M.H., Kadanoff, L.P. Procaccia, I. and Shraiman, B.I. (1986) Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A, 33, 1141-1151.
  • [7] Hentschel, H.G.E. and Procaccia, I. (1983) The infinite number of generalized dimensions of fractals and strange attractors. Physica D, 8, 435-444.
  • [8] Holley, R. and Waymire, E.C. (1992) Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl. Probab., 2, 819-845.
  • [9] Kahane, J.P. and Peyrière, J. (1976) Sur certaines martingales de Benoit Mandelbrot. Adv. Math., 22, 131-145.
  • [10] Lovejoy, S. and Schertzer, D. (1990) Multifractals, universality classes, and satellite and radar measurements of cloud and rain fields. J. Geophys. Res., 95, 2021-2034.
  • [11] Mandelbrot, B.B. (1974) Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech., 62, 331-358.
  • [12] Marani, M., Rinaldo, A., Rigon, R. and Rodriguez-Iturbe, I. (1994) Geomorphological width functions and the random cascade. Geophys. Res. Lett., 21, 2123-2126.
  • [13] Meneveau, C. and Chabra, A.B. (1990) Two-point statistics of multifractal measures. Physica A, 164, 564-574.
  • [14] Meneveau, C. and Sreenivasan, K.R. (1991) The multifractal nature of turbulent energy dissipation. J. Fluid Mech., 224, 429-484.
  • [15] Novikov, E.A. and Stewart, R.W. (1964) Intermittency of turbulence and the spectrum of fluctuations of energy dissipation. Izv. Akad. Nauk USSR, Geophys. Ser., 3, 408-413.
  • [16] Olsson, J. (1995) Limits and characteristics of the multifractal behavior of a high-resolution rainfall time series. Nonlinear Processes Geophys., 2, 23-29.
  • [17] Over, T.M. and Gupta, V.K. (1994) Statistical analysis of mesoscale rainfall: dependence of a random cascade generator on large-scale forcing. J. Appl. Meteorol., 33, 1526-1542.
  • [18] Platt, D.E. and Family, F. (1993) Consistent scaling of multifractal measures: multifractal spatial correlations. Phys. Rev. E, 47, 2281-2288.
  • [19] Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. New York: Wiley.
  • [20] She, Z.-S. and Waymire, E.C. (1995) Quantized energy cascade and log-Poisson statistics in fully developed turbulence. Phys. Rev. Lett., 74, 262-265.