On multivariate fractional random fields: tempering and operator-stable laws

Abstract

In this paper, we define a new and broad family of vector-valued random fields called tempered operator fractional operator-stable random fields (TRF, for short). TRF is typically non-Gaussian and generalizes tempered fractional stable stochastic processes. TRF comprises moving average and harmonizable-type subclasses that are constructed by tempering (matrix-) homogeneous, matrix-valued kernels in time-and Fourier-domain stochastic integrals with respect to vector-valued, strictly operator-stable random measures. We establish the existence and fundamental properties of TRF. Assuming both Gaussianity and isotropy, we show the equivalence between certain moving average and harmonizable subclasses of TRF. In addition, we establish sample path properties in the scalar-valued case for several Gaussian instances.

Multivariate tempered stable random fields

“…Then, taking into account the different notation in [6], it is easy to verify that we obtain the following observation as a by-product of Theorem 3.1 in [6] (namely for B = 1 α I). The details are left to the reader.…”

“…Recall from the proof of Theorem 2.5 in [14] that, in the context of Proposition 4.2 (a), condition H < β is needed in order to control the behavior of f t (s) from (3.3) for large values of s. Hence, as already announced before, this condition disappears when tempering in the sense of [6]. For the same reason we observe in Example 4.3 that ϕ has not to be admissible anymore.…”

“…Actually, the random fields from Section 3 have a true tempered stable character that arises from Proposition 2.5. Let us be more specific what we mean by the word true: Recall the main results from [6] and note that there a different kind of tempering is considered, which extends the univariate idea from [16]. That is, the authors in [6] somehow add a tempering function to the actual kernel function, say f (•), while they abstain from the use of T αS random measured as integrator.…”

“…On the one hand, it follows that the behavior of f (s) is restrained for large values of s. However, on the other hand, the resulting stochastic integrals have no T αS distribution. Also note that, in [6], the aforementioned approach has been performed both for a movingaverage representation and for a harmonizable representation. We focus on the first one.…”

“…Prominent examples are fractional stable processes given by either a moving-average or a harmonizable representation, see [22] and the literature cited there. By modifying the kernel function of fractional stable processes, so-called tempered fractional stable motions are obtained, see [6]. The purpose of this paper is to construct fractional tempered stable fields and processes and to study their basic properties.…”