  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 nontrivial statistically self-similar 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.
