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

DAW, Lara; LOOSVELDT, Laurent

doi:10.1214/22-EJP878

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

DAW, Lara; LOOSVELDT, Laurent

2022 • In Electronic Journal of Probability, 27, p. 1-45

Peer Reviewed verified by ORBi

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https://hdl.handle.net/10993/50566

DOI
10.1214/22-EJP878

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Keywords :

Wiener chaos,; Rosenblatt process,; Wavelet series,; Random Series,; slow/ordinary/rapid points,; modulus of continuity

Abstract :

[en] 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.

Disciplines :

Mathematics

Author, co-author :

DAW, Lara ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

LOOSVELDT, Laurent ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

External co-authors :

Language :

English

Title :

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

Publication date :

November 2022

Journal title :

Electronic Journal of Probability

eISSN :

1083-6489

Publisher :

Institute of Mathematical Statistics, Beachwood, United States - Ohio

Volume :

Pages :

1-45

Peer reviewed :

Peer Reviewed verified by ORBi

FnR Project :

FNR12582675 - Approximation Of Gaussian Functionals, 2018 (01/09/2019-31/08/2022) - Ivan Nourdin

Funders :

FNR - Fonds National de la Recherche

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since 11 March 2022

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Bibliography

J. M. P. Albin. A note on Rosenblatt distributions. Statist. Probab. Lett., 40(1):83–91, 1998. MR1650532
A. Arneodo, Y. d’Aubenton Carafa, E. Bacry, P.V. Graves, J.F. Muzy, and C. Thermes. Wavelet based fractal analysis of DNA sequences. Physica D.: Nonlinear Phenomena, (96):291–320, 1996. MR1368691
A. Arneodo, J.-F. Muzy, and E. Bacry. Wavelets and multifractal formalism for singular signals: application to turbulence data. Physical Review Letters, (67):3515–3518, 1991.
A. Ayache. Multifractional stochastic fields. World Scientific Publishing Co. Pte. Ltd., Hacken-sack, NJ, 2019. Wavelet strategies in multifractional frameworks. MR3839281
A. Ayache. Lower bound for local oscillations of Hermite processes. Stoch. Process. Their Appl., 130:4593–4607, 2020. MR4108464
A. Ayache and Y. Esmili. Wavelet-type expansion of the generalized Rosenblatt process and its rate of convergence. J. Fourier Anal. Appl., 26(3):Paper No. 51, 35, 2020. MR4110623
A. Ayache, C. Esser, and T. Kleyntssens. Different possible behaviors of wavelet leaders of the Brownian motion. Statist. Probab. Lett., 150:54–60, 2019. MR3922488
J.-M. Bardet and C. A. Tudor. A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter. Stochastic Process. Appl., 120(12):2331–2362, 2010. MR2728168
J. Barral and S. Seuret. From multifractal measures to multifractal wavelet series. J. Fourier Anal Appl., (11):589–614, 2005. MR2182637
F. Bastin, C. Esser, and S. Jaffard. Large deviation spectra based on wavelet leaders. Rev. Mat. Iberoam., 32(3):859–890, 2016. MR3556054
A. Bunde, J. Kropp, and H.J. Schellnhuber. The science of disasters: climate disruptions, heart attacks, and market crashes, volume 2. Springer-Verlag, 2002.
Amit Chaurasia. Performance of synthetic Rosenblatt process under multicore architecture. In 2019 3rd International conference on Electronics, Communication and Aerospace Technology (ICECA), pages 377–381. IEEE, 2019.
M. Clausel and S. Nicolay. Wavelet techniques for pointwise anti-Hölderian irregularity. Constr. Approx., 33:41–75, 2011. MR2747056
I. Daubechies. Orthonormal bases of compactly supported wavelets. Comm. Pure App. Math., 41:909–996, 1988. MR0951745
I. Daubechies. Ten lectures on wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, 1992. MR1162107
A. Deliège, T. Kleyntssens, and S. Nicolay. Mars topography investigated through the wavelet leaders method: a multidimensional study of its fractal structure. Planet. Space Sci., 136:46– 58, 2017.
R.L. Dobrushin and P. Major. Non-central limit theorems for nonlinear functionals of gaussian fields. Z. Wahrsch. Verw. Gebiete, (50):27–52, 1979. MR0550122
C. Esser and L. Loosveldt. Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions. To appear in ALEA, 2022.
P. Flandrin. Time-frequency / Time-scale analysis. In Wavelet Analysis and its Applications. Academic Press, 1999. MR1681043
S. Jaffard. Multifractal formalism for functions part I: Results valid for all functions. SIAM J. Math. Anal., (28):944–970, 1997. MR1453315
S. Jaffard. Wavelet techniques in multifractal analysis, fractal geometry and applications: A jubilee of Benoit Mandelbrot. Proceedings of Symposia in Pure Mathematics, 72:91–151, 2004. MR2112122
S. Jaffard, P. Abry, and S. Roux. Function spaces vs. Scaling functions: Some issues inimage classification. Mathematical Image processing, pages 1–40, 2011. MR2867518
S. Jaffard and Y. Meyer. Wavelet methods for pointwise regularity and local oscillations of functions. Mem. Amer. Math. Soc., 123(587):x+110, 1996. MR1342019
S. Janson. Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics. Cambridge University Press, 1997. MR1474726
J.-P. Kahane. Some random series of functions, volume 5 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1985. MR0833073
G. Kerchev, I. Nourdin, E. Saksman, and L. Viitasaari. Local times and sample path properties of the Rosenblatt process. Stoch. Process. Their Appl., 131:498–522, 2021. MR4165649
D. Kreit and S. Nicolay. Generalized pointwise Hölder spaces definitionined via admissible sequences. J. Funct. Spaces, ID 8276258:11, 2018. MR3812831
E. H. Lakhel and A. Tlidi. Existence, uniqueness and stability of impulsive stochastic neutral functional differential equations driven by rosenblatt process with varying-time delays. Random Operators and Stochastic Equations, 27(4):213–223, 2019. MR4036646
P.G. Lemarié and Y. Meyer. Ondelettes et bases hilbertiennes. Rev. Mat. Iberoam., 2:1–18, 1986. MR0864650
P. Lévy. Processus stochastiques et mouvement brownien. Gauthier-Villars, Paris, 1948. MR0190953
L. Loosveldt and S. Nicolay. Generalized spaces of pointwise regularity: Toward a general framework for the WLM. Nonlinearity, 34:6561–6586, 2021. MR4304490
M. Maejima and C.A. Tudor. Selfsimilar processes with stationnary increments in the second Wiener chaos. Probab. Math. Stat., 32(1):167–186, 2012. MR2959876
M. Maejima and C.A. Tudor. On the distribution of the Rosenblatt process. Statist. Probab. Lett., 83(6):1490–1495, 2013. MR3048314
S. Mallat. A Wavelet Tour of Signal Processing. Academic Press, 1999. MR1614527
Y. Meyer and D. Salinger. Wavelets and operators, volume 1. Cambridge university press, 1995. MR1228209
Y. Meyer, F. Sellan, and M. S. Taqqu. Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion. J. Fourier Anal. Appl., 5(5):465–494, 1999. MR1755100
S. Nicolay, M. Touchon, B. Audit, Y. d’Aubenton Carafa, C. Thermes, A. Arneodo, et al. Bifractality of human DNA strand-asymmetry profiles results from transcription. Phys. Rev. E, 75:032902, 2007.
Bernt Oksendal. Stochastic differential equations: an introduction with applications. Springer Science & Business Media, 2013. MR2001996
M. Rosenblatt. A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. U.S.A., 42:43–47, 1956. MR0074711
R. Sakthivel, P. Revathi, Y. Ren, and G. Shen. Retarded stochastic differential equations with infinite delay driven by rosenblatt process. Stochastic analysis and applications, 36(2):304– 323, 2018. MR3750683
L. Schwartz. Théorie des distributions. Hermann, 1978. MR0209834
M.S. Taqqu. The rosenblatt process. In Richard A. Davis, Keh-Shin Lii, and Dimitris N. Politis, editors, Selected Works of Murray Rosenblatt, pages 29–45. Springer New York, New York, NY, 2011. MR2742596
C.A. Tudor and F.G. Viens. Variations and estimators for self-similarity parameters via Malli-avin calculus. Ann. Probab., 37(6):2093–2134, 2009. MR2573552
M.S.. Veillette and M. S. Taqqu. Properties and numerical evaluation of the Rosenblatt distribution. Bernoulli, 19(3):982–1005, 2013. MR3079303
H. Wendt, P. Abry, S. Jaffard, H. Ji, and Z. Shen. Wavelet leader multifractal analysis for texture classification. ICIP, pages 3829–3832, 2009.
H. Wendt, S. Roux, S. Jaffard, and P. Abry. Wavelet leaders and bootstrap for multifractal analysis of images. Signal Processing, (89(6)):1100–1114, 2009.
N. Wiener. Collected works, volume 1. The MIT press, 1976. MR0532698
N. Wiener and R.C. Paley. Fourier transforms in the complex domain. Amer. Math. Soc. Colloq. Pub., 19, 1934. 183pp. MR1451142