  Multifractal analysis of multiplicative random cascades is revisited within
the framework of {\em 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 $\chi$ respectively above and beyond 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. 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.
