A simple construction of the fractional Brownian motion

Enriquez, Nathanaël

doi:10.1016/j.spa.2003.10.008

Cited by 61 publications

(72 citation statements)

References 14 publications

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“…In this class, each model deals with a specific underlying generic signal X -e.g., on-off processes (34), renewal processes (36,37), persistent random walks (38), and Ornstein-Uhlenbeck processes (39). The model established in this research is fundamentally different of the aforementioned superposition model: considering the randomization (via random transmission amplitudes and frequencies) of the superimposed signals rather than their stochastic scaling limits; considering arbitrary underlying generic signal processes rather than a specific one; and seeking amplitudinaluniversality and temporal-universality rather than setting as goal to obtain a fractional Brownian noise scaling limit.…”

Section: Section 4: Discussionmentioning

confidence: 99%

A unified and universal explanation for Lévy laws and 1/f noises

Eliazar

Klafter

2009

Proc. Natl. Acad. Sci. U.S.A.

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Lévy laws and 1/f noises are shown to emerge uniquely and universally from a general model of systems which superimpose the transmissions of many independent stochastic signals. The signals are considered to follow, statistically, a common-yet arbitrarygeneric signal pattern which may be either stationary or dissipative. Each signal is considered to have its own random transmission amplitude and frequency. We characterize the amplitudefrequency randomizations which render the system output's stationary law and power-spectrum universal-i.e., independent of the underlying generic signal pattern. The classes of universal stationary laws and power spectra are shown to coincide, respectively, with the classes of Lévy laws and 1/f noises-thus providing a unified and universal explanation for the ubiquity of these classes of "anomalous statistics" in various fields of science and engineering.anomalous statistics | universality | Poissonian randomizations | shot noise A nomalous statistics are ubiquitously observed in many fields of science and engineering, and their exploration has drawn major interest in recent years by both experimentalists and theoreticians (1-3). In the context of stationary stochastic processes and signals, one can observe either amplitudinal or temporal anomalous statistics.Amplitudinal anomalous statistics are manifested by wide process fluctuations, and are referred to as the Noah effect (4). Quantitatively, amplitudinal anomalous statistics are characterized by "heavy-tailed" stationary laws (5)-stationary laws whose distribution tails follow asymptotic power-law decay. The quintessential proxy of amplitudinal anomalous statistics is the class of Lévy laws (6, 7). This class of probability laws emerges from the Central Limit Theorem as the universal scaling limits of sums of independent and identically distributed (IID) random variables with infinite variance (8, 9).Temporal anomalous statistics are manifested by long process memory and are referred to as the Joseph effect (4). Quantitatively, temporal anomalous statistics are characterized by longrange correlations (10, 11)-autocorrelation functions following asymptotic power-law decay. The quintessential proxy of temporal anomalous statistics is the class of 1/f noises (12-14)-stationary processes with power-law power spectra.Stationary stochastic processes and signals are prevalent across many fields of science and engineering. Examples of such processes and signals include the intensity of solar luminosity, the sales of a consumer product, and transmissions sent through communication channels. In many systems, a very large collection of microscopic stationary stochastic inputs are aggregated up to form a macroscopic stationary system output. In the context of the aforementioned examples, consider, respectively, the luminosity of a galaxy, the sales of a large department store, and transmissions sent through a central communication router.In this research, we consider systems whose outputs are superpositions of many stationary stochastic in...

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Section: Section 4: Discussionmentioning

confidence: 99%

A unified and universal explanation for Lévy laws and 1/f noises

Eliazar

Klafter

2009

Proc. Natl. Acad. Sci. U.S.A.

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“…For example, Mandelbort and Salvatore applied fractional calculus to processing white noises and images and acquired some desirable results (Enriquez et al, 2004;Salvatore and Mario, 2014). In China, Yao and Zhou conducted in-depth studies on the characteristics of the fractional calculus image for the fractal function.…”

Section: Reconstruction Of Fractal Modelmentioning

confidence: 99%

Reconstruction of surface errors of the metallic waveguide based on electromagnetic scattering theory

Tang

2016

IJSURFSE

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Abstract:First, the fractal model of surface errors was constructed by means of the Monte Carlo method, and the source term and spectral amplitude of surface errors were calculated. The reconstruction model of surface errors was then established. The non-differentiable problem of the fractal model was solved by the fractional differential theory. To verify the accuracy of proposed models, a waveguide filter was manufactured. The reconstruction results obtained by the fractal function model were compared with the measured results, and a comparison of the results obtained by different reconstruction models proves that the fractal function is optimal in modelling.Keywords: reconstruction; surface error; metallic waveguide; electromagnetic scattering; fractal function.Reference to this paper should be made as follows: Li, N. and Tang, B. (2016) 'Reconstruction of surface errors of the metallic waveguide based on electromagnetic scattering theory', Int.

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“…Most of them give rise to approximate syntheses, such as the midpoint displacement method (see [23], for instance), the wavelet based decomposition ( [1,22,27], etc. ), or more recently a method based on correlated random walks [14]. A few of them can be applied not only for 1-dimensional fBm but also to simulate 2-dimensional (anisotropic) fractional Brownian fields and lead to approximate syntheses.…”

Section: Simulationmentioning

confidence: 99%

Estimation of anisotropic Gaussian fields through Radon transform

Biermé¹,

Richard²

2007

ESAIM: PS

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Abstract.We estimate the anisotropic index of an anisotropic fractional Brownian field. For all directions, we give a convergent estimator of the value of the anisotropic index in this direction, based on generalized quadratic variations. We also prove a central limit theorem. First we present a result of identification that relies on the asymptotic behavior of the spectral density of a process. Then, we define Radon transforms of the anisotropic fractional Brownian field and prove that these processes admit a spectral density satisfying the previous assumptions. Finally we use simulated fields to test the proposed estimator in different anisotropic and isotropic cases. Results show that the estimator behaves similarly in all cases and is able to detect anisotropy quite accurately.Mathematics Subject Classification. 60G60,62M40,60G15,60G10,60G17,60G35,44A12.

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A simple construction of the fractional Brownian motion

Cited by 61 publications

References 14 publications

A unified and universal explanation for Lévy laws and 1/f noises

A unified and universal explanation for Lévy laws and 1/f noises

Reconstruction of surface errors of the metallic waveguide based on electromagnetic scattering theory

Estimation of anisotropic Gaussian fields through Radon transform

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