Anisotropic scaling of the random grain model with application to network traffic

Pilipauskaitė, Vytautė; Сургайлис, Донатас

doi:10.1017/jpr.2016.45

Cited by 18 publications

(22 citation statements)

References 22 publications

(100 reference statements)

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“…have a similar structure and properties to generalized Hermite processes discussed in [3] except that (3.7) are defined as k-tuple Itô-Wiener integrals with respect to white noise in R 2 and not in R as in [3]. Following the terminology in [28], RFs xZ + k (y) and yZ − k (x) may be called a generalized Hermite slide since they represent a random surface 'sliding linearly to 0' along one of the coordinate on the plane from a generalized Hermite process indexed by the other coordinate. In the Gaussian case k = 1, a…”

Section: )mentioning

confidence: 99%

“…The last fact together with Proposition 5.1 (i) implies the following corollary. Remark 6.1 Following the terminology in [28], we say that a covariance stationary RF X = {X(t, s), (t, s) ∈ Z 2 } has vertical LRD property (respectively, horizontal LRD property) if s∈Z |r X (0, s)| = ∞ (respectively, t∈Z |r X (t, 0)| = ∞). From Corollary 6.1 we see the dichotomy of the limit distribution in Theorems 3.2 -3.3 at points kp 2 = 1 at kp 1 = 1 is related to the change of vertical and horizontal LRD properties of the subordinated RF X = A k (Y ).…”

Section: Properties Of Convolutions Of Generalized Homogeneous Functionsmentioning

confidence: 99%

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Scaling transition for nonlinear random fields with long-range dependence

Pilipauskaitė

Сургайлис

2017

Stochastic Processes and their Applications

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Section: )mentioning

confidence: 99%

Section: Properties Of Convolutions Of Generalized Homogeneous Functionsmentioning

confidence: 99%

Scaling transition for nonlinear random fields with long-range dependence

Pilipauskaitė

Сургайлис

2017

Stochastic Processes and their Applications

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“…We mention that random germ-grain models have received significant attention in the literature (cf. [3]- [8], [14], [18], [23], and [25]). In a nutshell, our paper extends the work from Biermé et al [4] and Kaj et al [18] to a random boxes model where the size of a grain depends on two differently heavy-tailed-distributed random variables instead of just one random variable for the volume of the grain.…”

Section: Modelmentioning

confidence: 99%

“…An anisotropic scaling was examined by Pilipauskaitė and Surgailis [25]. They studied the scaling limits of the random grain model on the plane with heavy-tailed grain area distribution.…”

Section: Related Workmentioning

confidence: 99%

Scaling limits for a random boxes model

Аурзада

Schwinn

2019

Adv. Appl. Probab.

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We consider random rectangles in R 2 that are distributed according to a Poisson random measure, i.e., independently and uniformly scattered in the plane. The distributions of the length and the width of the rectangles are heavy-tailed with different parameters. We investigate the scaling behaviour of the related random fields as the intensity of the random measure grows to infinity while the mean edge lengths tend to zero. We characterise the arising scaling regimes, identify the limiting random fields and give statistical properties of these limits.2010 Mathematics Subject Classification: 60G60 (primary); 60F05, 60G55 (secondary).Keywords: Gaussian random field, generalised random field, Poisson point process, Poisson random field, random balls model, random grain model, random field, stable random field.Since our purpose is to deal with centred random fields, we introduce the notation for the corresponding centred Poisson random measure N := N − n and centred integral J(µ) := J(µ) − EJ(µ).The goal of this paper is to obtain scaling limits for the random field J. By scaling, we mean that the length and the width of the boxes are shrinking to zero, i.e., the scaled edge lengths are ρu i with scaling parameter ρ → 0, and that the expected number of boxes is increasing, i.e., the intensity λ of the Poisson point process is tending to infinity as a function of ρ. The precise behaviour of λ = λ(ρ) → ∞ is specified in the different scaling regimes below. Following the notational convention from above, we denote by J ρ the centred random field corresponding to the Poisson random measure N ρ with the modified intensity λ ρ := λ(ρ) and scaled edge lengths, i.e., F ρ is the image measure of F by the change u → ρu.Next, we want to say a few words about the applications of random balls models. The motivation comes from models from telecommunication networks. A list of some references can be found at the beginning of Chapter 3

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“…For 'cross-sectional' product fields X v (u)X v+s (u + t), (u, v) ∈ Z 2 , s = 0 involving cross-sectional lags, a similar scaling transition occurs in the interval 0 < β < 3/2 with the critical rate N ∼ n 2β between different scaling regimes, see Theorem 3.1. The notion of scaling transition for longrange dependent random fields in Z 2 was discussed in Puplinskaitė and Surgailis [23], [24], Pilipauskaitė and Surgailis [19], [20].…”

Section: Introductionmentioning

confidence: 99%

Sample covariances of random-coefficient AR(1) panel model

Leipus¹,

Philippe²,

Pilipauskaitė³

et al. 2019

Electron. J. Statist.

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The present paper obtains a complete description of the limit distributions of sample covariances in N × n panel data when N and n jointly increase, possibly at different rate. The panel is formed by N independent samples of length n from random-coefficient AR(1) process with the tail distribution function of the random coefficient regularly varying at the unit root with exponent β > 0. We show that for β ∈ (0, 2) the sample covariances may display a variety of stable and non-stable limit behaviors with stability parameter depending on β and the mutual increase rate of N and n.

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Anisotropic scaling of the random grain model with application to network traffic

Cited by 18 publications

References 22 publications

Scaling transition for nonlinear random fields with long-range dependence

Scaling transition for nonlinear random fields with long-range dependence

Scaling limits for a random boxes model

Sample covariances of random-coefficient AR(1) panel model

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