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Multifractal analysis of sums of random pulses

Part of: Classical measure theory Real functions

Published online by Cambridge University Press:  22 June 2023

GUILLAUME SAËS  and
STÉPHANE SEURET
GUILLAUME SAËS
Affiliation:
Laboratoire d’Analyse et de Mathématiques appliquées, Université Paris-Est Créteil, Créteil, France. e-mails: guillaume.saes@u-pec.fr, seuret@u-pec.fr
STÉPHANE SEURET
Affiliation:
Laboratoire d’Analyse et de Mathématiques appliquées, Université Paris-Est Créteil, Créteil, France. e-mails: guillaume.saes@u-pec.fr, seuret@u-pec.fr
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Abstract

In this paper, we investigate the regularity properties and determine the almost sure multifractal spectrum of a class of random functions constructed as sums of pulses with random dilations and translations. In addition, the continuity moduli of the sample paths of these stochastic processes are investigated.

MSC classification

Primary: 28A80: Fractals 28A78: Hausdorff and packing measures 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.)

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 175 , Issue 3 , November 2023 , pp. 569 - 593
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

REFERENCES

Abry, P., Jaffard, S. and Wendt, H.. Irregularities and scaling in signal and image processing: multifractal analysis. In Benoit Mandelbrot: A Life in Many Dimensions, M. Frame and N. Cohen, eds. (World Scientific Publishing 2015), pp 31–116.CrossRefGoogle Scholar
Ahsanullah, T. M. and Nevzorov, V. B.. Probability Theory (Springer, 2015)CrossRefGoogle Scholar
Amo, E., Bhouri, I. and Fernàndez–Sànchez, J.. A note on the Hausdorff dimension of general sums of pulses graphs. Rend. Circ. Mat. Palermo 60 (2011), 110–123.Google Scholar
Aubry, J.-M. and Jaffard, S.. Random wavelet series. Comm. Math. Phys. Rend. Circ. Mat. Palermo (2002), 483–514.CrossRefGoogle Scholar
Barral, J., Fournier, N., Jaffard, S. and Seuret, S.. A pure jump Markov process with a random singularity spectrum. Ann. Probab. 38(5) (2010), 1924–1946.Google Scholar
Barral, J. and Mandelbrot, B.. Multifractal products of cylindrical pulses. Probab. Theory Related Fields 124 (1985), 125–147.Google Scholar
Barral, J. and Seuret, S.. Random sparse sampling in a Gibbs weighted tree. J. Inst. Math. Jussieu 19(1) (2020), 65–116.CrossRefGoogle Scholar
Barral, J. and Seuret, S.. A localised Jarnik–Besicovich theorem. Adv. Math. 226 (2011), 3191–3215.CrossRefGoogle Scholar
Bayart, F.. Multifractal spectra of typical and prevalent measures. Nonlinearity 26(2) (2013), 353–367.CrossRefGoogle Scholar
Ben Abid, M.. Existence and Hölder regularity of pulse functions. Colloq. Math. 116(2) (2009), 217–225.CrossRefGoogle Scholar
Beresnevitch, V. and Velani, S.. A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. Math. 164(3) (2009), 971–992.Google Scholar
Bertoin, J.. Lévy Processes. (Cambridge University Press, 1998).Google Scholar
Breiman, L.. Probability. Society for Industrial and Applied Mathematics (1992).CrossRefGoogle Scholar
Buczolich, Z. and Nagy, J.. Hölder spectrum of typical monotone continuous functions. Real Anal. Exchange 26(1) (1999), 133–156.Google Scholar
Buczolich, Z. and Seuret, S.. Typical Borel measures on [0, 1]d satisfy a multifractal formalism. Nonlinearity 23(11) (2010), 2905–2918.CrossRefGoogle Scholar
Cioczek–Georges, R., Mandelbrot, B., Samorodnitsky, G. and Taqqu, M. S.. Stable fractal sums of pulses: the cylindrical case. Bernoulli 1(3) (1995), 201–216.CrossRefGoogle Scholar
Cioczek–Georges, R. and Mandelbrot, B.. A class of micropulses and antipersistent fractional brownian motion. Stochastic Process. Appl. 60(1) (1995), 1–18.CrossRefGoogle Scholar
Cioczek–Georges, R. and Mandelbrot, B.. Alternative micropulses and fractional brownian motion. Stochastic Process. Appl. 64(2) (1996), 143–152.CrossRefGoogle Scholar
Demichel, Y.. Analyse fractale d’une famille de fonctions aléatoire: les fonctions de bosses. PhD. thesis. Université Blaise Pascal (France) (2006).Google Scholar
Demichel, Y. and Tricot, C.. Analysis of the fractal sum of pulses. Math. Proc. Camb. Phil. Soc. 141(2) (2006), 355–370.CrossRefGoogle Scholar
Demichel, Y. and Falconer, K.. The Hausdorff dimension of pulse-sum graphs. Math. Proc. Camb. Phil. Soc. 143(1) (2007), 143–145.Google Scholar
Dodson, M. M., Melian, M. V., Pestana, P. and Velani, S. L.. Patterson measure and ubiquity. Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 37–60.Google Scholar
Falconer, K.. Fractal Geometry (Wiley, 1990).Google Scholar
Frisch, U. and Parisi, G.. On the singularity structure of fully developed turbulence. Turbulence and predictability in geophysical fluid dynamics and climate dynamics (1974), 84–88.Google Scholar
Gagne, Y.. Etude expérimentale de l’intermittence et des singularités dans le plan complexe en turbulence pleinement développée. PhD. thesis. Université Joseph Fourier (France) (1987).Google Scholar
Bordenave, C., Gousseau, Y. and Roueff, F.. The dead leaves model: an example of a general tesselation. Adv. Appl. Probability 38(1) (2006), 31–46.CrossRefGoogle Scholar
Calka, P. and Demichel, Y.. Fractal random series generated by Poisson-Voronoi tessellations Trans. Amer. Math. Soc. 367(6) (2015), 4157–4182.CrossRefGoogle Scholar
Heurteaux, Y.. Weierstrass functions with random phases Trans. Amer. Math. Soc. 355(8) (2003), 3065–3077.Google Scholar
Hunt, B.R.. The Hausdorff dimension of graphs of Weierstrass functions. Proc. Amer. Math. Soc. 126(3) (1998), 781–800.CrossRefGoogle Scholar
Jaffard, S.. Exposants de Hölder en des points donnés et coefficients d’ondelettes. C.R. Acad. Sci. Paris Sér. I Math. Number 4 308 (1989), 79–82.Google Scholar
Jaffard, S.. The multifractal nature of Lévy processes. Probab. Theory Related Fields 114(2) (1999), 207–227.Google Scholar
Jaffard, S.. On the Frisch–Parisi conjecture. J. Math. Pures Appl. 79(6) (2000), 525–552.CrossRefGoogle Scholar
Jaffard, S.. On lacunary wavelet series. Ann. Appl. Probab. 10(1) (2000), 313–329.Google Scholar
Jaffard, S. and Martin, B.. Multifractal analysis of the Brjuno function. Invent. Math. 212 (2018), 109–132.CrossRefGoogle Scholar
Jaffard, S., Melot, C., Leonarduzzi, R., Wendt, H., Abry, P., Roux, S. and Torres, M.. p-exponent and p-leaders, Part I: Negative pointwise regularity. Phys. A 448 (2016), 300–318.Google Scholar
Jaffard, S. and Meyer, Y.. Wavelet methods for pointwise regularity and local oscillations of functions. Amer. Math. Soc. 123 (1996).CrossRefGoogle Scholar
Kahane, J.-P.. Some Random Series of Function (Cambridge University Press, 1985).Google Scholar
Khoshnevisan, D., Xiao, Y. and Zhong, Y.. Measuring the range of an additive Lévy process. Ann. Probab. 31(2) (2003), 1097–1141.CrossRefGoogle Scholar
Lovejoy, S. and Mandelbrot, B.. Fractal properties of rain, and a fractal model. Tellus 37(3) (1985), 209–232.CrossRefGoogle Scholar
Mandelbrot, B.. Introduction to fractal sums of pulses In: Levy Flights and Related Topics in Physics 450 (1995). M. F. Shlesinger, G. M. Zaslavsky and U. Frisch (Eds.), (Springer Berlin Heidelberg publishing), 110–123.Google Scholar
Mattila, P.. Fourier Analysis and Hausdorff Dimension (Cambridge University Press, 2015).CrossRefGoogle Scholar
Pesin, Y.. The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples. Chaos 7(89) (1997), 89–106.Google Scholar
Roueff, F.. Dimension de Hausdorff du graphe d’une fonction continue: une étude analytique et statistique. PhD. thesis. Ecole Nationale Supérieure des Télécommunications (2000).Google Scholar
Shepp, L. A.. Covering the circle with random ares. Israel J. Math. 11 (1972), 328–345.CrossRefGoogle Scholar
Shieh, N.-R. and Xiao, Y.. Hausdorff and packing dimensions of the images of random fields. Bernoulli 16(4) (2010), 926–952.CrossRefGoogle Scholar