Localizable Moving Average Symmetric Stable and Multistable Processes

Falconer, K. J.; Guével, Ronan Le; Véhel, Jacques Lévy

doi:10.1080/15326340903291321

Cited by 27 publications

(36 citation statements)

References 8 publications

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“…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. There the stability index α : R → R + can vary with time t. Based on random integrals of deterministic functions so called multi-stable processes were introduced.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

Multi operator-stable random measures and fields

Kremer

Scheffler

2019

Stochastic Models

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“…We call a stochastic process {Y (t), t ∈ R} multistable if for almost all u, Y is localisable at u with Y u an α-stable process for some α = α(u), where 0 < α(u) ≤ 2. Various constructions of multistable processes are given in [8,9,11].…”

Section: Multistable Processes and Localisabilitymentioning

confidence: 99%

“…Some of these are considered in [8,9,11] using alternative definitions of multistable processes. It is convenient to make the convention that…”

Section: Examplesmentioning

confidence: 99%

Multistable Processes and Localizability

Falconer

Liu

2012

Stochastic Models

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We use characteristic functions to construct α-multistable measures and integrals, where the measures behave locally like stable measures, but with the stability index α(x) varying with x. This enables us to construct α-multistable processes on R, that is processes whose scaling limit at time t is an α(t)-stable process. We present several examples of such multistable processes and examine their localisability.

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“…MBm was introduced independently in [PLV95] and [BJR97] and since then there is an increasing interest in the study of multifractional processes, we refer for instance to [FaL,S08] for two excellent quite recent articles on this topic. The main three features of mBm are the following:…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Recent Developments in Fractals and Related Fields

Barral

Seuret

2010

Applied and Numerical Harmonic Analysis

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: Multifractional Brownian motion (mBm), denoted here by X, is one of the paradigmatic examples of a continuous Gaussian process whose pointwise Hölder exponent depends on the location. Recall that X can be obtained (see e.g. [BJR97, AT05]) by replacing the constant Hurst parameter H in the standard wavelet series representation of fractional Brownian motion (fBm) by a smooth function H(·) depending on the time variable t. Another natural idea (see [BBCI00]) which allows to construct a continuous Gaussian process, denoted by Z, whose pointwise Hölder exponent does not remain constant all along its trajectory, consists in substituting H(k/2 j ) to H in each term of index (j, k) of the standard wavelet series representation of fBm. The main goal of our article is to show that X and Z only differ by a process R which is smoother than them; this means that they are very similar from a fractal geometry point of view.

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Localizable Moving Average Symmetric Stable and Multistable Processes

Cited by 27 publications

References 8 publications

Multi operator-stable random measures and fields

Multi operator-stable random measures and fields

Multistable Processes and Localizability

Recent Developments in Fractals and Related Fields

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