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Probability density of the empirical wavelet coefficients of a noisy chaos

Author

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  • Matthieu Garcin

    (Natixis Asset Management - SAMS, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Dominique Guegan

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

Abstract

We are interested in the random empirical wavelet coefficients of a noisy signal when this signal is a unidimensional or multidimensional chaos. More precisely we provide an expression of the conditional probability density of such coefficients, given a discrete observation grid. The noise is assumed to be described by a symmetric alpha-stable random variable. If the noise is a dynamic noise, then we present the exact expression of the probability density of each wavelet coefficient of the noisy signal. If we face a measurement noise, then the noise has a non-linear influence and we propose two approximations. The first one relies on a Taylor expansion whereas the second one, relying on an Edgeworth expansion, improves the first general Taylor approximation if the cumulants of the noise are defined. We give some illustrations of these theoretical results for the logistic map, the tent map and a multidimensional chaos, the Hénon map, disrupted by a Gaussian or a Cauchy noise.

Suggested Citation

  • Matthieu Garcin & Dominique Guegan, 2014. "Probability density of the empirical wavelet coefficients of a noisy chaos," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01310473, HAL.
    • Handle: RePEc:hal:cesptp:hal-01310473
    DOI: 10.1016/j.physd.2014.03.005
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    Cited by:

    1. Matthieu Garcin, 2019. "Fractal analysis of the multifractality of foreign exchange rates [Analyse fractale de la multifractalité des taux de change]," Working Papers hal-02283915, HAL.
    2. Matthieu Garcin & Jules Klein & Sana Laaribi, 2022. "Estimation of time-varying kernel densities and chronology of the impact of COVID-19 on financial markets," Working Papers hal-02901988, HAL.
    3. Ammy-Driss, Ayoub & Garcin, Matthieu, 2023. "Efficiency of the financial markets during the COVID-19 crisis: Time-varying parameters of fractional stable dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).

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