Sample Path Properties of Operator-Slef-Similar Gaussian Random Fields

Mason, J. David; Xiao, Yimin

doi:10.1137/s0040585x97978749

Cited by 40 publications

(36 citation statements)

References 19 publications

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“…The framework of operator self-similar (o.s.s.) random processes and fields was originally conceived by Laha and Rohatgi (1981), Hudson and Mason (1982), and has attracted much attention recently (e.g., Maejima and Mason (1994), Mason and Xiao (2002), Biermé et al (2007), Xiao (2009), Guo et al (2009), Pipiras (2011, 2012), Vedel (2011, 2013), Li and Xiao (2011), Dogan et al (2014), Puplinskaitė and Surgailis (2015), Didier et al (2017aDidier et al ( , 2017b). If H = diag(h 1 , .…”

Section: Introductionmentioning

confidence: 99%

Wavelet eigenvalue regression for n-variate operator fractional Brownian motion

Abry

Didier

2018

Journal of Multivariate Analysis

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In this contribution, we extend the methodology proposed in Abry and Didier (2017a) to obtain the first joint estimator of the real parts of the Hurst eigenvalues of n-variate OFBM. The procedure consists of a wavelet regression on the log-eigenvalues of the sample wavelet spectrum. The estimator is shown to be consistent for any time reversible OFBM and, under stronger assumptions, also asymptotically normal starting from either continuous or discrete time measurements. Simulation studies establish the finite sample effectiveness of the methodology and illustrate its benefits compared to univariate-like (entrywise) analysis. As an application, we revisit the well-known self-similar character of Internet traffic by applying the proposed methodology to 4-variate time series of modern, high quality Internet traffic data. The analysis reveals the presence of a rich multivariate self-similarity structure.

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Section: Introductionmentioning

confidence: 99%

Wavelet eigenvalue regression for n-variate operator fractional Brownian motion

Abry

Didier

2018

Journal of Multivariate Analysis

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“…In this paper we provide a brief reanalysis of the data collected by the latter researchers. Techniques have been developed that can handle this anisotropy in the scaling or the Hurst coefficients [ Dobrushin , 1978; Hudson and Mason , 1982; Schertzer and Lovejoy , 1985, 1987; Kumar and Foufoula‐Georgiou , 1993; Mason and Xiao , 2001; Bonami and Estrade , 2003] yet the mathematical character of numerical implementations is still under investigation, since many of the properties of these fields have only recently been worked out in one dimension [ Pipiras and Taqqu , 2000, 2003].…”

Section: Introductionmentioning

confidence: 99%

Aquifer operator scaling and the effect on solute mixing and dispersion

Benson

Meerschaert

Baeumer

et al. 2006

Water Resources Research

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[1] Since aquifer parameters may have statistical dependence structures that are present across a huge range of scales, the concepts of fractional Brownian motion (fBm) have been used in both analytic and numerical stochastic settings. Most previous models have used isotropic scaling characterized by a single scalar Hurst coefficient. Any real-world anisotropy has been handled by an elliptical stretching random K field. We define a d-dimensional extension of fBm in which the fractional-order integration may take on different orders in the d primary (possibly nonorthogonal) scaling directions, and the degree of connectivity and long-range dependence is freely assigned via a probability measure on the unit sphere. This approach accounts for the different scaling found in the vertical versus horizontal directions in sedimentary aquifers and allows very general degrees of continuity of K in certain directions. It also allows for the representation of fracture networks in a continuum setting: The eigenvectors of the scaling matrix describe the primary fracture scaling directions, and discrete weights of fractional integration represent fracture continuity that may be limited to a small number of directions. In a numerical experiment, the motion of solutes through 2-D ''operator-fractional'' Gaussian fields depends very much on transverse Hurst coefficients. Transverse scaling in the range of fractional Gaussian noise engenders greater plume mixing and a transition to a Fickian regime. Higher orders of integration, in the range of fractional Brownian motion, are associated with thicker layering of aquifer sediments and more preferential, unmixed transport. Therefore direct representation of the unique directional scaling properties of an aquifer is important for realistic simulation of transport.

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“…. < a p for the distinct real parts of the eigenvalues of E. This includes the case of operator fractional Brownian motions studied in [36,12,13] and operator scaling stable random fields [4], where corresponding Hausdorff dimension results already were already established in [36,4,5]. Note that, since the real parts of the eigenvalues of E and D are assumed to be positive, the matrices c E and c D are contracting for any 0 < c < 1.…”

Section: 1mentioning

confidence: 99%

On the carrying dimension of occupation measures for self-affine random fields

Kern¹,

Sönmez²

2019

pms

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Hausdorff dimension results are a classical topic in the study of path properties of random fields. This article presents an alternative approach to Hausdorff dimension results for the sample functions of a large class of self-affine random fields. We present a close relationship between the carrying dimension of the corresponding self-affine random occupation measure introduced by U. Zähle and the Hausdorff dimension of the graph of self-affine fields. In the case of exponential scaling operators, the dimension formula can be explicitly calculated by means of the singular value function. This also enables to get a lower bound for the Hausdorff dimension of the range of general self-affine random fields under mild regularity assumptions. d = X(t − s) for all s, t ∈ R d , where " d =" denotes equality in distribution.

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Sample Path Properties of Operator-Slef-Similar Gaussian Random Fields

Cited by 40 publications

References 19 publications

Wavelet eigenvalue regression for n-variate operator fractional Brownian motion

Wavelet eigenvalue regression for n-variate operator fractional Brownian motion

Aquifer operator scaling and the effect on solute mixing and dispersion

On the carrying dimension of occupation measures for self-affine random fields

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