Large deviations for multiplicative chaos

Collet, Pierre; Koukiou, F.

doi:10.1007/bf02096590

Cited by 68 publications

(44 citation statements)

References 14 publications

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“…Similar conclusions in different contexts were shown in [3], [18], [33], [37] for multiplicative cascades, [38] for Galton-Watson processes, and [21], [25] for branching processes in random environments.…”

Section: Resultssupporting

confidence: 70%

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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Section: Resultssupporting

confidence: 70%

Moments for multidimensional Mandelbrot cascades

Huang¹

2016

J. Appl. Probab.

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“…Multinomial measures, quasi-Bernoulli measures, Mandelbrot cascades and their extensions, as well as the recent compound Poisson cascades, are examples of such multiplicative chaos measures [14,40,33,17,7,8]. These measures typically have a multifractal spectrum with the wellknown ∩-shape, reflecting the validity of a multifractal formalism.…”

Section: Introductionmentioning

confidence: 99%

Combining Multifractal Additive and Multiplicative Chaos

Barral

Seuret

2005

Commun. Math. Phys.

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ABSTRACT. In this work, we study the new class of multifractal measures, which combines additive and multiplicative chaos, defined bywhere µ is any positive Borel measure on [0, 1] and b is an integer ≥ 2.The singularities analysis of the measures νγ,σ involves new results on the mass distribution of µ when µ describes large classes of multifractal measures. These results generalize ubiquity theorems associated with the Lebesgue measure. Under suitable assumptions on µ, the multifractal spectrum of νγ,σ is linear on [0, hγ,σ] for some critical value hγ,σ, and then it is strictly concave on the right of hγ,σ, and deduced from the one of µ by an affine transformation. This untypical shape is the result of the combination between Dirac masses and atomless multifractal measures. These measures satisfy multifractal formalisms. These measures open interesting perspectives in modeling discontinuous phenomena.

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“…These large deviation properties have a geometrical counterpart given by the multifractal analysis of µ as well as of its variants [37][38][39][40][41][42][43]. In the following theorem the cascade is no longer assumed to be canonical, but it is still assumed that the components of W do not vanish and that ϕ > −∞ on R (this imposes λ j > 0 for all j).…”

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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Large deviations for multiplicative chaos

Abstract: The local singularities for a class of random measures, obtained by random iterated multiplications, are investigated using the thermodynamic formalism. This analysis can be interpreted as a rigorous study of the phase transition of a system with random interactions.

Cited by 68 publications

References 14 publications

Moments for multidimensional Mandelbrot cascades

Moments for multidimensional Mandelbrot cascades

Combining Multifractal Additive and Multiplicative Chaos

Mandelbrot's cascades: a legendary destiny

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