The interest for multifractal stochastic processes is mainly motivated by the need for accurate models in the study of the variability of wild signals. These locally irregular signals come from physical phenomena such as fully developed turbulence, TCP Internet traffic, variations of financial prices, or heart beats.

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 that 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.

Abstract. The familiar cascade measures are sequences of random positive measures obtained on [0,1] via b-adic independent cascades. To generalize them, this paper allows the random weights invoked in the cascades to take real or complex values. This yields sequences of random functions whose possible strong or weak limits are natural candidates for modeling multifractal phenomena. Their asymptotic behavior is investigated, yielding a sufficient condition for almost sure uniform convergence to non-trivial statistically selfsimilar limits. Is the limit function a monofractal function in multifractal time? General sufficient conditions are given under which such is the case, as well as examples for which no natural time change can be used. In most cases when the sufficient condition for convergence does not hold, we show that either the limit is 0 or the sequence diverges almost surely. In the later case, a functional central limit theorem holds, under some conditions. It provides a natural normalization making the sequence converge in law to a standard Brownian motion in multifractal time. 1.