Operator-stable and operator-self-similar random fields

Kremer, Dustin; Scheffler, Hans–Peter

doi:10.1016/j.spa.2018.11.013

Cited by 13 publications

(36 citation statements)

References 17 publications

(72 reference statements)

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“…Recall Remark 2.3 and that µ s ∼ [0, 0, ϕ(s, ·)] with ϕ(s, ·) being symmetric for every s ∈ S. Then (a) and (b) follow immediately from Proposition 3.3 and section 5 of [10], respectively. Also recall that µ s is full for every s ∈ S. Hence, using (2.13) and Lemma 1.3.11 in [14], a slight refinement of Proposition 2.6 (a) in [11] gives part (c). The details are left to the reader.…”

Section: Multi Operator-stable Random Measures and Integralsmentioning

confidence: 94%

“…In particular, if we assume that µ = µ s is α-stable for fixed 0 < α < 2, our construction of random measures covers the considerations in [2], [13] and [19]. Moreover, if µ = µ s is still constant but operator-stable (say with exponent B), we also get back the setting of [11]. Merely note that, in contrast to [11], µ has to be symmetric (instead of strictly) operator-stable in our framework.…”

Section: Multi Operator-stable Random Measures and Integralsmentioning

confidence: 99%

“…Using the general theory of ISRMs developed in [10], we now construct a large class of multivariate independently scattered random measures which are first of all infinitely-divisible. At the same time we refer the reader to [8], [11] and [14] concerning more details about operator-stable distributions which will play a crucial role throughout the paper. In order to start with some notation, let L(R m ) be the set of all linear operators on R m and GL(R m ) the corresponding subset of all invertible operators, represented as m × mmatrices in each case.…”

Section: Multi Operator-stable Random Measures and Integralsmentioning

confidence: 99%

“…For a comprehensive introduction see [14]. Based on the recently developed theory of multivariate independently scattered random measures (short: ISRMs) and their integrals (see [10]), moving-average and harmonizable representations of (E, D)-operator-self-similar random fields with operator-stable marginals are presented in [11]. Our main feature of SαS ISRMs or more generally of operator-stable ISRMs with exponent B (a m × m-matrix) is that they are homogeneous in the time variable t ∈ R d , that is α or B are constant and do not depend on t. Motivated by various applications, in [6] scalarvalued so called multi-stable random measures were constructed.…”

Section: Introductionmentioning

confidence: 99%

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Multi operator-stable random measures and fields

Kremer

Scheffler

2019

Stochastic Models

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In this paper we construct vector-valued multi operator-stable random measures that behave locally like operator-stable random measures. The space of integrable functions is characterized in terms of a certain quasi-norm. Moreover, a multi operatorstable moving-average representation of a random field is presented which behaves locally like an operator-stable random field which is also operator-self-similar. 1 λ(s) := λ B(s) as well as Λ(s) := Λ B(s) and assume thatHence with a := inf s∈S Λ(s) −1 and b := sup s∈S λ(s) −1 we have for any s ∈ S thatRemark 2.1. Fix s ∈ S and assume that B(s) = diag(b 1 (s), ..., b m (s)) is diagonal. Then we get that r B(s) = diag(r b 1 (s) , ..., r bm(s) ) and the mapping (0, ∞) ∋ r → r B(s) x is strictly increasing for every x = 0. On the other hand, if B(s) is merely symmetric, there exists an orthogonal matrix O(s) ∈ GL(R m ) such that r B(s) = O(s)r D(s) O(s) * for some diagonal matrix D(s) with spec(D(s)) = spec(B(s)), see Proposition 2.2.2 in [14]. Hence the previous observation remains true, since x = 0. Overall and according to Lemma 6.1.5 in [14] this allows us to assume that · B(s) = · and that S B(s) = S m−1 = {x ∈ R m : x = 1} for any s ∈ S. Moreover, let σ be a finite and symmetric measure on (S m−1 , B(S m−1 )), where B(S m−1 ) denotes the corresponding collection of Borel sets. Then for every s ∈ S the mapping (2.2) ϕ(s, C) := ∞ 0 S m−1

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Section: Multi Operator-stable Random Measures and Integralsmentioning

confidence: 94%

Section: Multi Operator-stable Random Measures and Integralsmentioning

confidence: 99%

Section: Multi Operator-stable Random Measures and Integralsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multi operator-stable random measures and fields

Kremer

Scheffler

2019

Stochastic Models

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“…26) where d ∈ (−M, M ). Thus, combining (2.25) with (2.26) and the triangle inequality, one gets (2.24).…”

mentioning

confidence: 99%

Wavelet series representation and geometric properties of harmonizable fractional stable sheets

2019

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Let Z H = {Z H (t), t ∈ R N } be a real-valued N -parameter harmonizable fractional stable sheet with index H = (H 1 , . . . , H N ) ∈ (0, 1) N . We establish a random wavelet series expansion for Z H which is almost surely convergent in all the Hölder spaces C γ ([−M, M ] N ), where M > 0 and γ ∈ (0, min{H 1 , . . . , H N }) are arbitrary. One of the main ingredients for proving the latter result is the LePage representation for a rotationally invariant stable random measure.Also, let X = {X(t), t ∈ R N } be an R d -valued harmonizable fractional stable sheet whose components are independent copies of Z H . By making essential use of the regularity of its local times, we prove that, on an event of positive probability, the formula for the Hausdorff dimension of the inverse image X −1 (F ) holds for all Borel sets F ⊆ R d . This is referred to as a uniform Hausdorff dimension result for the inverse images.

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