Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process

Daw, Lara; Loosveldt, Laurent

doi:10.1214/22-ejp878

Cited by 6 publications

(3 citation statements)

References 40 publications

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“…Also, with this article, we hope to open the door to future development of simulation methods for such generalized chaotic processes for which no simulation method is available so far. We hope as well to open the door to that of new strategies allowing to study in depth their erratic local sample path behavior, as for instance to show the existence of slow points and rapid points, in the same spirit of what has been very recently done for FBM in [12] and for generalized Rosenblatt process in [9].…”

Section: Introductionmentioning

confidence: 69%

Wavelet-Type Expansion of Generalized Hermite Processes with rate of convergence

Ayache¹,

Hamonier²,

Loosveldt³

2023

Preprint

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Wavelet-type random series representations of the well-known Fractional Brownian Motion (FBM) and many other related stochastic processes and fields have started to be introduced since more than two decades. Such representations provide natural frameworks for approximating almost surely and uniformly rough sample paths at different scales and for study of various aspects of their complex erratic behavior.Hermite process of an arbitrary integer order d, which extends FBM, is a paradigmatic example of a stochastic process belonging to the dth Wiener chaos. It was introduced very long time ago, yet many of its properties are still unknown when d ≥ 3. In a paper published in 2004, Pipiras raised the problem to know whether wavelet-type random series representations with a well-localized smooth scaling function, reminiscent to those for FBM due to Meyer, Sellan and Taqqu, can be obtained for a Hermite process of any order d. He solved it in this same paper in the particular case d = 2 in which the Hermite process is called the Rosenblatt process. Yet, the problem remains unsolved in the general case d ≥ 3. The main goal of our article is to solve it, not only for usual Hermite processes but also for generalizations of them. Another important goal of our article is to derive almost sure uniform estimates of the errors related with approximations of such processes by scaling functions parts of their wavelet-type random series representations.

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Section: Introductionmentioning

confidence: 69%

Wavelet-Type Expansion of Generalized Hermite Processes with rate of convergence

Ayache¹,

Hamonier²,

Loosveldt³

2023

Preprint

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“…Hölder regularity only compare the oscillations with power functions while, with moduli of continuity, one can deduce more precise and relevant information concerning the analysed function. It is particularly true when we consider stochastic processes, see for instance [31,27,30]. Note that one can define generalized Hölder spaces associated with modulus of continuity [39,40,46] and that these spaces lead to specific multifractal formalisms [47,48].…”

Section: Preliminaries Strategy and Main Resultsmentioning

confidence: 99%

“…On this purpose, we use a generalization of a combination of previous ideas from the papers [6,9,27]. First, remark that, similarly to the proof of Theorem 2.9, it suffices to show that, for all n ∈ N, there is Ω n , an event of probability 1, such that, on Ω n , for all t 0 ∈ [n, n + 1], (25) holds.…”

Section: Pointwise Hölder Exponentmentioning

confidence: 99%

Multifractional Hermite processes: definition and first properties

Loosveldt¹

2023

Preprint

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We define multifractional Hermite processes which generalize and extend both multifractional Brownian motion and Hermite processes. It is done by substituting the Hurst parameter in the definition of Hermite processes as a multiple Wiener-Itô integral by a Hurst function. Then, we study the pointwise regularity of these processes, their local asymptotic self-similarity and some fractal dimensions of their graph. Our results show that the fundamental properties of multifractional Hermite processes are, as desired, governed by the Hurst function. Complements are given in the second order Wiener chaos, using facts from Malliavin calculus.

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Wavelet-Type Expansion of Generalized Hermite Processes with Rate of Convergence

Ayache,

Hamonier,

Loosveldt

2024

Constr Approx

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Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process

Cited by 6 publications

References 40 publications

Wavelet-Type Expansion of Generalized Hermite Processes with rate of convergence

Wavelet-Type Expansion of Generalized Hermite Processes with rate of convergence

Multifractional Hermite processes: definition and first properties

Wavelet-Type Expansion of Generalized Hermite Processes with Rate of Convergence

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