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Multifractional Hermite processes: definition and first properties Loosveldt, Laurent E-print/Working paper (2023) 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 39 (1 UL)Wavelet-Type Expansion of Generalized Hermite Processes with rate of convergence ; ; Loosveldt, Laurent E-print/Working paper (2023) 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 44 (1 UL)On the pointwise regularity of the Multifractional Brownian Motion and some extensions ; Loosveldt, Laurent E-print/Working paper (2023) We study the pointwise regularity of the Multifractional Brownian Motion and in particular, we get the existence of slow points. It shows that a non self-similar process can still enjoy this property. We ... [more ▼] We study the pointwise regularity of the Multifractional Brownian Motion and in particular, we get the existence of slow points. It shows that a non self-similar process can still enjoy this property. We also consider various extensions of our results in the aim of requesting a weaker regularity assumption for the Hurst function without altering the regularity of the process. [less ▲] Detailed reference viewed: 19 (2 UL)Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process Daw, Lara ; Loosveldt, Laurent in Electronic Journal of Probability (2022), 27 We identify three types of pointwise behaviour in the regularity of the (generalized) Rosenblatt process. This extends to a non Gaussian setting previous results known for the (fractional) Brownian motion ... [more ▼] We identify three types of pointwise behaviour in the regularity of the (generalized) Rosenblatt process. This extends to a non Gaussian setting previous results known for the (fractional) Brownian motion. On this purpose, fine bounds on the increments of the Rosenblatt process are needed. Our analysis is essentially based on various wavelet methods. [less ▲] Detailed reference viewed: 78 (19 UL)Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions ; Loosveldt, Laurent in ALEA: Latin American Journal of Probability and Mathematical Statistics (2022), 19 We study the Hölderian regularity of Gaussian wavelets series and show that they display, almost surely, three types of points: slow, ordinary and rapid. In particular, this fact holds for the Fractional ... [more ▼] We study the Hölderian regularity of Gaussian wavelets series and show that they display, almost surely, three types of points: slow, ordinary and rapid. In particular, this fact holds for the Fractional Brownian Motion. Finally, we remark that the existence of slow points is specific to these functions. [less ▲] Detailed reference viewed: 23 (1 UL)Some Prevalent Sets in Multifractal Analysis: How Smooth is Almost Every Function in T_p^\alpha(x) Loosveldt, Laurent ; in Journal of Fourier Analysis and Applications (2022), 28(4), We present prevalent results concerning generalized versions of the $T_p^\alpha$ spaces, initially introduced by Calderón and Zygmund. We notably show that the logarithmic correction appearing in the ... [more ▼] We present prevalent results concerning generalized versions of the $T_p^\alpha$ spaces, initially introduced by Calderón and Zygmund. We notably show that the logarithmic correction appearing in the quasi-characterization of such spaces is mandatory for almost every function; it is in particular true for the Hölder spaces, for which the existence of the correction was showed necessary for a specific function. We also show that almost every function from $T_p^α (x0 )$ has α as generalized Hölder exponent at $x_0$. [less ▲] Detailed reference viewed: 35 (5 UL) |
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