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

Abstract

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

Operator-stable and operator-self-similar random fields

“…Then (5.10) also holds with Re Z j in the numerator and Re Z j is a modification of Re Z ′ j as well. This allows to finish the proof quite similar to Proposition 4.1 in [19] as well as to Proposition 5.1 in [3], because Y j and Re Z ′ j have the same finite-dimensional distributions. To verify this we fix 1 ≤ j ≤ m, t 1 , ..., t n ∈ R d and u 1 , ..., u n ∈ R for some n ∈ N. Then we obtain with Proposition 5.4 and Remark 5.12 in [11] that For the additional statement of this theorem merely use (5.9) and write τ E (s − t) β j − ε 2 = τ E (s − t) β j −ε τ E (s − t) ε 2 since the growth of the logarithm is more slowly than that of any polynomial with positive degree.…”

“…Similarly in the situation of (ii) we merely have to verify that φ(s) Re λ j + −ε φ(s) −D * e j is bounded for s ≥ L. For this purpose the definition of j + and D * = diag(D * 1 , ..., D * p ) obviously permit a slight extension of Corollary 2.2 (b) in [19], leading to φ(s) −D * e j ≤ C ′ 0 φ(s) −Re λ j + +ε , s ≥ L for a corresponding constant C ′ 0 .…”

“…(b) A combination of Theorem 5.6 and the results in [8] allows to calculate the Hausdorff dimension of the image and the graph of the harmonizable representation, at least for the present case. Then we should get back the results of Theorem 5.1 in [19] which will be independent of B again, i.e. they only depend on spec(E) and spec(D).…”

“…Finally we want to consider a special case which is still purely operator-stable and which is inspired by the α-stable case, considered in [3] and [19]. Hence for the rest of this section we assume that the assumptions of Theorem 5.1 are fulfilled.…”

Sample Path Properties of Generalized Random Sheets with Operator Scaling

“…, whereas in higher space dimensions all indices are relevant and the corresponding results in more general dimensions can be found in [18]. See also [19,20] for related results.…”