Self-similar Random Fields and Rescaled Random Balls Models

Biermé, Hermine; Estrade, Anne; Kaj, Ingemar

doi:10.1007/s10959-009-0259-x

Cited by 35 publications

(78 citation statements)

References 20 publications

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“…This correspondence helps us to unify and extend results on asymptotic fluctuations of heavy-tailed sources under aggregation, and to clarify the role in this context of fractional Poisson motion [2,5] and of fractional Brownian motion as rescaling limit processes.…”

Section: Introduction and Main Resultsmentioning

confidence: 75%

“…By analogy, the mean zero non-Gaussian process P H has been called fractional Poisson motion, c.f. [2,5]. By a refined analysis of higher-order moments it can be shown moreover that P H and B H share the same Hölder-regularity: in both cases the paths are γ-Hölder continuous of any order γ < H, [6].…”

Section: Theorem 14mentioning

confidence: 91%

“…Recall that β and are given in (2) and that, in the scaling (ICR), the sequence a = a m satisfies a → ∞ and m (a)/a β → µc β as m → ∞. With b(a, m) = a, Theorem 1.1 implies, for k ≥ 1,…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

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Moment measures of heavy-tailed renewal point processes: asymptotics and applications

Dombry

Kaj

2013

ESAIM: PS

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PostprintThis is the accepted version of a paper published in ESAIM. P&S. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination. Abstract. We study higher-order moment measures of heavy-tailed renewal models, including a renewal point process with heavy-tailed inter-renewal distribution and its continuous analog, the occupation measure of a heavy-tailed Lévy subordinator. Our results reveal that the asymptotic structure of such moment measures are given by explicit power-law density functions. The same power-law densities appear naturally as cumulant measures of certain Poisson and Gaussian stochastic integrals. This correspondence provides new and extended results regarding the asymptotic fluctuations of heavy-tailed sources under aggregation, and clarifies existing links between renewal models and fractional random processes.

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Section: Introduction and Main Resultsmentioning

confidence: 75%

Section: Theorem 14mentioning

confidence: 91%

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Moment measures of heavy-tailed renewal point processes: asymptotics and applications

Dombry

Kaj

2013

ESAIM: PS

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“…The limiting operations are carried out with the use of generalized random fields based on a careful choice of the space of measures M. The results in generalize the Gaussian, stable and intermediate limits obtained here to a spatial setting and are in complete analogy to those of Theorems 1, 2 and 3, for the case of fixed rewards. Biermé et al (2006Biermé et al ( , 2007, extend the Gaussian and the intermediate scaling limit results further for an analogous model where the intensity of the size of grains has a specified power law behavior close to zero. It turns out that for such models one can obtain in the scaling limit, for example, the family of fractional Brownian fields {B H (x), x ∈ R d } with Hurst index H, 0 < H < 1.…”

Section: Remaining Choices For the Parameters γ And δmentioning

confidence: 53%

“…Some of the approximation techniques we use have been unified in Pipiras and Taqqu (2006). In addition, our approach for the case of fixed rewards has been successfully extended in a spatial setting of Poisson germ-grain models and recast in a more abstract formulation involving random fields in , further developed in Biermé et al (2006Biermé et al ( , 2007.…”

Section: Introductionmentioning

confidence: 99%

In and Out of Equilibrium 2

Sidoravičius¹,

Vares²

2008

Progress in Probability

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This is an author-produced version of a chapter published in In and Out of Equilibrium 2. It does not include the final publisher proof-corrections or pagination. Abstract. It has become common practice to use heavy-tailed distributions in order to describe the variations in time and space of network traffic workloads. The asymptotic behavior of these workloads is complex; different limit processes emerge depending on the specifics of the work arrival structure and the nature of the asymptotic scaling. We focus on two variants of the infinite source Poisson model and provide a coherent and unified presentation of the scaling theory by using integral representations. This allows us to understand physically why the various limit processes arise.

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Introduction to Random Fields and Scale Invariance

Biermé

2019

Stochastic Geometry

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In medical imaging, several authors have proposed to characterize roughness of observed textures by their fractal dimensions. Fractal analysis of 1D signals is mainly based on the stochastic modeling using the famous fractional Brownian motion for which the fractal dimension is determined by its so-called Hurst parameter. Lots of 2D generalizations of this toy model may be defined according to the scope. This lecture intends to present some of them. After an introduction to random fields, the first part will focus on the construction of Gaussian random fields with prescribed invariance properties such as stationarity, self-similarity, or operator scaling property. Sample paths properties such as modulus of continuity and Hausdorff dimension of graphs will be settled in the second part to understand links with fractal analysis. The third part will concern some methods of simulation and estimation for these random fields in a discrete setting. Some applications in medical imaging will be presented. Finally, the last part will be devoted to geometric constructions involving Marked Poisson Point Processes and shot noise processes. Definition 2. The distribution of (X t) t∈T is given by all its finite dimensional distribution (fdd) ie the distribution of all real random vectors (X t 1 ,. .. , X t k) for k ≥ 1,t 1 ,. .. ,t k ∈ T. Note that joint distributions for random vectors of arbitrary size k are often difficult to compute. However we can infer some statistics of order one and two by considering only couples of variables. Definition 3. The stochastic process (X t) t∈T is a second order process if E(X 2 t) < + ∞, for all t ∈ T. In this case we define • its mean function m X : t ∈ T → E(X t) ∈ R; • its covariance function K X : (t, s) ∈ T × T → Cov(X t , X s) ∈ R. We have therefore a very simple condition on variogram for Gaussian random fields with stationary increments. Corollary 2. Let X be a centered Gaussian field with stationary increments. If for all ε > 0, there exist c 1 , c 2 > 0, such that for all x ∈ [−1, 1] d , c 1 x 2γ+ε ≤ v X (x) = E((X x − X 0) 2) ≤ c 2 x 2γ−ε , then X has critical Hölder exponent on [0, 1] d equal to γ.

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Self-similar Random Fields and Rescaled Random Balls Models

Cited by 35 publications

References 20 publications

Moment measures of heavy-tailed renewal point processes: asymptotics and applications

Moment measures of heavy-tailed renewal point processes: asymptotics and applications

In and Out of Equilibrium 2

Introduction to Random Fields and Scale Invariance

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