MULTIFRACTAL ANALYSIS IN A MIXED ASYMPTOTIC FRAMEWORK

• 成果类型: Article
• 署名作者: Bacry, Emmanuel; Gloter, Arnaud; Hoffmann, Marc; Muzy, Jean Francois
• 署名单位: Institut Polytechnique de Paris; ENSTA Paris; Ecole Polytechnique; Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Institut Polytechnique de Paris; ENSAE Paris; Centre National de la Recherche Scientifique (CNRS); CNRS - Institute for Engineering & Systems Sciences (INSIS)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/09-AAP670
• 发表日期: 2010
• 页码: 1729-1760
• 关键词: large deviations; cascades; distributions; dimensions; martingales; turbulence; Markov; LAW

摘要

Multifractal analysis of multiplicative random cascades is revisited within the framework of mixed asymptotics. In this new framework, the observed process can be modeled by a concatenation of independent binary cascades and statistics are estimated over a sample whose size increases as the resolution scale (or the sampling period) becomes finer. This allows one to continuously interpolate between the situation where one studies a single cascade sample at arbitrary fine scales and where, at fixed scale, the sample length (number of cascades realizations) becomes infinite. We show that scaling exponents of mixed partitions functions, that is, the estimator of the cumulant generating function of the cascade generator distribution depends on some mixed asymptotic exponent chi, respectively, above and below two critical value p(chi)(-) and p(chi)(+). We study the convergence properties of partition functions in mixed asymtotics regime and establish a central limit theorem. Moreover, within the mixed asymptotic framework, we establish a boxcounting multifractal formalism that can be seen as a rigorous formulation of Mandelbrot's negative dimension theory. Numerical illustrations of our results on specific examples are also provided. A possible application of these results is to distinguish data generated by log-Normal or log-Poisson models.