Multifractal analysis in a mixed asymptotic framework

Abstract

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 χ, respectively, above and below 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. 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 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.