Multivariate stochastic integrals with respect to independently scattered random measures on δ-rings

Kremer, Dustin; Scheffler, Hans–Peter

doi:10.48550/arxiv.1711.00890

Cited by 4 publications

(40 citation statements)

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“…If we write • q×q for the operator norm in R q×q then, clearly, g t (x) 2p×2p ≤ 2 max{ ℜ g t (x) p×p , ℑ g t (x) p×p }. Hence, the conditions of Theorem 2.5 in Kremer and Scheffler (2019) are satisfied, which implies the process (3.22) exists due to Proposition 5.10 in Kremer and Scheffler (2017). Moreover, by Corollary 5.11(b) in Kremer and Scheffler (2017), the characteristic function of the candidate limiting process (3.22) at times t 1 , .…”

Section: B Proofs: Sectionmentioning

confidence: 68%

“…(A.12)) exists, and the Lévy symbol ψ satisfies ψ(c B u) = cψ(u), i.e., the function e ψ(u) is the characteristic function of a strictly operator-stable distribution ν B (see Kremer and Scheffler (2019), p. 4085). Based on ν B , we define the random measures L B used in Proposition 3.3 as R p -valued (in (3.19)) or C p -valued (in (3.22)) infinitely divisible independently scattered random measures on (R, B(R)) generated by ν B and the Lebesgue measure on R in the sense of Example 3.7(a) and Remark 3.8 in Kremer and Scheffler (2017).…”

mentioning

confidence: 99%

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On operator fractional Lévy motion: integral representations and time reversibility

Boniece¹,

Didier²

2021

Preprint

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In this paper, we construct operator fractional Lévy motion (ofLm), a broad class of non-Gaussian stochastic processes that are covariance operator self-similar, have wide-sense stationary increments and display infinitely divisible marginal distributions. The ofLm class generalizes the univariate fractional Lévy motion as well as the multivariate operator fractional Brownian motion (ofBm). The ofLm class can be divided into two types, namely, moving average (maofLm) and real harmonizable (rhofLm), both of which share the covariance structure of ofBm under assumptions. We show that maofLm and rhofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small-and large-scale limiting behavior. We characterize time reversibility for ofLm through parametric conditions related to its Lévy measure, starting from a framework for the uniqueness of finite second moment, multivariate stochastic integral representations. In particular, we show that, under non-Gaussianity, the parametric conditions for time reversibility are generally more restrictive than those for the Gaussian case (ofBm).

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Section: B Proofs: Sectionmentioning

confidence: 68%

mentioning

confidence: 99%

On operator fractional Lévy motion: integral representations and time reversibility

Boniece¹,

Didier²

2021

Preprint

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“…, u m ∈ R n and r > 0. By Kremer and Scheffler (2017), Theorem 5.4, (b), and Example 3.7, (a) (see also expression (2.23) in this paper), the characteristic function of the vector (X λ (x 1 ), . .…”

Section: Appendix a Section 3: Proofsmentioning

confidence: 81%

“…We now show (b). By Theorem 5.4, (c), in Kremer and Scheffler (2017), after a change of variables it suffices to show that…”

Section: Appendix a Section 3: Proofsmentioning

confidence: 99%

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

Didier

Kanamori

Sabzikar

2020

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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.

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