A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter

• Published in: Stochastic analysis and applications Vol. 120; no. 12; pp. 2331 - 2362
• Main Authors: Bardet, J.-M., Tudor, C.A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.12.2010; Elsevier; Taylor & Francis: STM, Behavioural Science and Public Health Titles
• Series: Stochastic Processes and their Applications
• Subjects: Distribution theory; Exact sciences and technology; Fractional Brownian motion; Limit theorems; Mathematics; Multiple Wiener-Ito integral Wavelet analysis Rosenblatt process Fractional Brownian motion Noncentral limit theorem Self-similarity Parameter estimation; Multiple Wiener-Itô integral; Noncentral limit theorem; Parameter estimation; Probability; Probability and statistics; Probability theory and stochastic processes; Rosenblatt process; Sciences and techniques of general use; Self-similarity; Statistics; Statistics Theory; Stochastic analysis; Stochastic processes; Wavelet analysis; secondary; Noncentral limit theorem; Parameter estimation; Rosenblatt process; Multiple Wiener-Itô integral; Wavelet analysis; Fractional Brownian motion; primary; Self-similarity; Chaos; Fractional brownian motion; Stochastic motion; Stochastic process; Wiener Itô integral; Stochastic integral; Limit theorem; Multiple integral; Mathematical expansion; Expansion; Self-similar process; secondary 60F05; 60H05; 62F12; Statistical moment; Asymptotic behavior; primary 60G18; Selfsimilarity; Statistical method; Gaussian process; Central limit theorem; Simulation; Wavelet transformation; Gaussian processes; wavelet analysis; parameter estimation; fractional Brownian motion; self-similarity; noncentral limit theorem
• ISSN: 0304-4149; 0736-2994; 1879-209X; 1532-9356
• DOI: 10.1016/j.spa.2010.08.003

By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we illustrate our results by simulations.