Multifractal Analysis of Complex Random Cascades

Barral, Julien; Jin, Xiong

doi:10.1007/s00220-010-1030-y

Cited by 25 publications

(35 citation statements)

References 58 publications

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“…The function p → ζ(p, s ) given in (16), extended on the entire space of real numbers, satisfies (17) and (21), the function p → ζ f (p, s ) is concave and independent on orthonormal wavelet bases in the Schwartz class. It also satisfies (22). Let us now show the second result; let s 2 = αs 1 + (1 − α)s 3 with 0 < α < 1.…”

Section: Scaling Functionmentioning

confidence: 86%

“…Other definitions of random wavelet series have been the subject of many papers. [21][22][23][24] In Sec. 2, we first recall the definition of oscillation spaces, then we extend ζ(p, s ) to p ∈ R. We show that it is concave with respect to p ∈ R and independent on orthonormal wavelet bases in the Schwartz class (using some results from Jaffard et al 25 ) and we prove its concavity with respect to s when p > 0, see Proposition 1.…”

Section: Multifractal Formalism Of Oscillating Singularities For Randmentioning

confidence: 99%

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Multifractal Formalism of Oscillating Singularities for Random Wavelet Series

Slimane

Halouani

2015

Fractals

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The oscillating multifractal formalism is a formula conjectured by Jaffard expected to yield the spectrum d(h, β) of oscillating singularity exponents from a scaling function ζ(p, s ), for p > 0 and s ∈ R, based on wavelet leaders of fractional primitives f −s of f . In this paper, using some results from Jaffard et al., we first show that ζ(p, s ) can be extended on p ∈ R to a function that is concave with respect to p ∈ R and independent on orthonormal wavelet bases in the Schwartz class. We also establish its concavity with respect to s when p > 0. Then, we prove that, under some assumptions, the extended scaling function ζ(p, s ) is the Legendre transform of the wavelet leaders density of f −s . Finally, as an application, we study the validity of the extended oscillating multifractal formalism for random wavelet series (under the assumption of independence and laws depending only on the scale).

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Section: Scaling Functionmentioning

confidence: 86%

Section: Multifractal Formalism Of Oscillating Singularities For Randmentioning

confidence: 99%

Multifractal Formalism of Oscillating Singularities for Random Wavelet Series

Slimane

Halouani

2015

Fractals

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“…Related work can be found in [4] and [5]. Here we are still interested in the existence of the αth-moment (α > 1) of the solution.…”

Section: Moments For the Complex Casementioning

confidence: 98%

Moments for multidimensional Mandelbrot cascades

Huang¹

2016

J. Appl. Probab.

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We consider the distributional equationwhere N is a random variable taking value in N 0 = {0, 1, . . .}, A 1 , A 2 , . . . are p × p nonnegative random matrices, and Z, Z(1), Z(2), . . . , are independent and identically distributed random vectors in R p + with R + = [0, ∞), which are independent of (N, A 1 , A 2 , . . .). Let {Y n } be the multidimensional Mandelbrot martingale defined as sums of products of random matrices indexed by nodes of a Galton-Watson tree plus an appropriate vector. Its limit Y is a solution of the equation above. For α > 1, we show a sufficient condition for E Y α ∈ (0, ∞). Then for a nondegenerate solution Z of the distributional equation above, we show the decay rates of Ee −t·Z as t → ∞ and those of the tail probability P(y ·Z ≤ x) as x → 0 for given y = (y 1 , . . . , y p ) ∈ R p + , and the existence of the harmonic moments of y ·Z. As an application, these results concerning the moments (of positive and negative orders) of Y are applied to a special multitype branching random walk. Moreover, for the case where all the vectors and matrices of the equation above are complex, a sufficient condition for the L α convergence and the αth-moment of the Mandelbrot martingale {Y n } are also established.

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“…The precise theorem can be found in [45], where a finer multifractal analysis of F is described (also see [51,72,[80][81][82] to have a complete overview of the multifractal formalism for functions). It follows from this theorem that, for the limits of conservative critical signed cascades (Theorem 5.3.2), the multifractal spectrum of F , i.e., the map h → dim E F (h), vanishes at 0 and has an infinite right derivative at 0.…”

Section: Multifractal Analysis Of Roughness In the Graph Of Fmentioning

confidence: 99%

“…To obtain the upper bound dim E µ (α) ≤ lim →0 lim inf n→∞ f (α, n, ) is easy. The study of points ϕ (q − ) et ϕ (q + ), more delicate, is done in [42] and [45].…”

Section: Large Deviations and Multifractal Analysismentioning

confidence: 99%

Mandelbrot's cascades: a legendary destiny

Barral¹,

Peyrière²

2015

Benoit Mandelbrot

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The original density is 1 for t ∈ (0, 1), b is an integer base (b ≥ 2), and p ∈ (0, 1) is a parameter. The first construction stage divides the unit interval into b subintervals and multiplies the density in each subinterval by either 1 or −1 with the respective frequencies of 1 2 + p 2 and 1 2 − p 2 . It is shown that the resulting density can be renormalized so that, as n → ∞ (n being the number of iterations) the signed measure converges in some sense to a nondegenerate limit. If H = 1 + log b p > 1/2, hence p > b −1/2 , renormalization creates a martingale, the convergence is strong, and the limit shares the Hölder and Hausdorff properties of the fractional Brownian motion of exponent H. If H ≤ 1/2, hence p ≤ b −1/2 , this martingale does not converge. However, a different normalization can be applied, for H ≤ 1 2 to the martingale itself and for H > 1 2 to the discrepancy between the limit and a finite approximation. In all cases the resulting process is found to converge weakly to the Wiener Brownian motion, independently of H and of b. Thus, to the usual additive paths toward Wiener measure, this procedure adds an infinity of multiplicative paths.

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Multifractal Analysis of Complex Random Cascades

Cited by 25 publications

References 58 publications

Multifractal Formalism of Oscillating Singularities for Random Wavelet Series

Multifractal Formalism of Oscillating Singularities for Random Wavelet Series

Moments for multidimensional Mandelbrot cascades

Mandelbrot's cascades: a legendary destiny

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