Multifractal analysis in a mixed asymptotic framework

Emmanuel Bacry, Arnaud Gloter, Marc Hoffmann, Jean François Muzy
2010 The Annals of Applied Probability  
Multifractal analysis of multiplicative random cascades is revisited within the framework of mixed asymptotics. In this new framework, statistics are estimated over a sample which 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
more » ... scaling exponents of "mixed" partitions functions i.e., the estimator of the cumulant generating function of the cascade generator distribution, depends on some "mixed asymptotic" exponent χ respectively above and beyond two critical value p − χ and p + χ . We study the convergence properties of partition functions in mixed asymtotics regime and establish a central limit theorem. These results are shown to remain valid within a general wavelet analysis framework. Their interpretation in terms of Besov frontier are discussed. Moreover, within the mixed asymptotic framework, we establish a "box-counting" multifractal formalism that can be seen as a rigorous formulation of Mandelbrot's negative dimension theory. Numerical illustrations of our purpose on specific examples are also provided.
doi:10.1214/09-aap670 fatcat:wqq3fisxdzdyjmrkx6q7keeigm