The Hausdorff dimension of multivariate operator-self-similar Gaussian random fields

Sönmez, Ercan

doi:10.1016/j.spa.2017.05.003

Cited by 10 publications

(14 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In accordance with Corollary 3.7, a comparison with (3.12) directly shows that this value coincides with s(c D , c q ) for any c ∈ (0, 1), indicating that the lower bound in Theorem 3.6 is in fact equal to the Hausdorff dimension of the range for the harmonizable representation of any (E, D)-operator-self-similar stable random field. All the above results also hold for a moving average representation of the random field in the Gaussian case α = 2 as shown in [45]. However, for a corresponding moving average representation in the stable case α ∈ (0, 2), constructed in [33], it is questionable if our results are applicable, since these fields do not share the same Hölder continuity properties and thus the joint measurability of sample functions (assumption (iv) in the Introduction) may be violated; cf.…”

Section: 1supporting

confidence: 52%

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

Kern¹,

Sönmez²

2019

pms

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: 1supporting

confidence: 52%

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

Kern¹,

Sönmez²

2019

pms

Self Cite

View full text Add to dashboard Cite

show abstract

“…it is not supported on any proper hyperplane in R m . As noted above, Sönmez [33] studied the sample path properties of X α in the Gaussian case α = 2. We will derive similar results in this paper for X α with α ∈ (0, 2).…”

Section: Harmonizable Representationmentioning

confidence: 99%

“…Proposition 4.1 compared to [33,Proposition 4.6] shows that (E, D)-operator-selfsimilar stable random fields share the same kind of upper bound for the modulus of continuity as the Gaussian ones. Therefore it is natural to have also the same results of [33,Theorem 4.1] for the Hausdorff dimension of their images and graphs on [0, 1] d , which we state in the next Section. Furthermore, we refer the reader to [14,28] for the definition and properties of the Hausdorff dimension.…”

Section: Modulus Of Continuitymentioning

confidence: 99%

“…Lastly, they leave the open problem of investigating the sample path regularity and fractal dimensions of these fields. In particular, they conjecture that these properties such as path continuity and Hausdorff dimensions are mostly determined by the real parts of the eigenvalues of E and D. Sönmez [33] solved this problem for the moving-average and harmonizable representation in the Gaussian case α = 2 and generalized several results in the literature (see [6,27,24,40]). In particular, he highlighted that the Hausdorff dimension of the range and the graph over a sample path depends on the real parts of the eigenvalues of E and D as well as the multiplicity of the eigenvalues of E and D. The purpose of this paper is to establish the corresponding results in the stable case α ∈ (0, 2) for the harmonizable representation.…”

Section: Introductionmentioning

confidence: 96%

“…show that harmonizable α-stable operator-self-similar random fields have the same kind of regularity properties as Gaussian operator-self-similar random fields. We will do this by applying results from [8] and by generalizing arguments used in [33,40].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fractal Behavior of Multivariate Operator-self-similar Stable Random Fields

Sönmez

2017

COSA

Self Cite

View full text Add to dashboard Cite

We investigate the sample path regularity of multivariate operator-selfsimilar α-stable random fields with values in R m given by a harmonizable representation. Such fields were introduced in [25] as a generalization of both operatorself-similar stochastic processes and operator scaling random fields and satisfy the scaling propertyand D is a real m×m matrix. By using results in [8] we give an upper bound on the modulus of continuity. Based on this we determine the Hausdorff dimension of the sample paths. In particular, this solves an open problem in [25]. 1 2 ERCAN SÖNMEZ random fields [6,7]. Recall that a stochastic processwhereas a scalar valued random field {Y (t) : t ∈ R d } is said to be operator scaling of order E and some H > 0 ifNote that (E, D)-operator-self-similar random fields can be seen as an anisotropic generalization of an operator-self-similar random fieldThe theoretical importance of self-similar random fields has increased significantly during the past four decades. They are also useful to model various natural phenomena for instance in physics, geophysics, mathematical engineering, finance or internet traffic, see, e.g., [23,1,31,35,10,9,5,12,36]. A very important class of such fields is given by Gaussian random fields and, in particular, by fractional Brownian fields

show abstract

Sample Path Properties of Generalized Random Sheets with Operator Scaling

Sönmez

2020

J Theor Probab

Self Cite

View full text Add to dashboard Cite

We consider operator scaling $$\alpha $$ α -stable random sheets, which were introduced in Hoffmann (Operator scaling stable random sheets with application to binary mixtures. Dissertation Universität Siegen, 2011). The idea behind such fields is to combine the properties of operator scaling $$\alpha $$ α -stable random fields introduced in Biermé et al. (Stoch Proc Appl 117(3):312–332, 2007) and fractional Brownian sheets introduced in Kamont (Probab Math Stat 16:85–98, 1996). We establish a general uniform modulus of continuity of such fields in terms of the polar coordinates introduced in Biermé et al. (2007). Based on this, we determine the box-counting dimension and the Hausdorff dimension of the graph of a trajectory over a non-degenerate cube $$I \subset {\mathbb {R}}^d$$ I ⊂ R d .

show abstract

The Hausdorff dimension of multivariate operator-self-similar Gaussian random fields

Cited by 10 publications

References 38 publications

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

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

Fractal Behavior of Multivariate Operator-self-similar Stable Random Fields

Sample Path Properties of Generalized Random Sheets with Operator Scaling

Contact Info

Product

Resources

About