  In this paper we extend some classical results valid for canonical
multiplicative cascades to exact scaling log-infinitely divisible cascades. We
complete previous results on non-degeneracy and moments of positive orders
obtained by Barral and Mandelbrot, and Bacry and Muzy: we provide a necessary
and sufficient condition for the non-degeneracy of the limit measures of these
cascades, as well as for the finiteness of moments of positive orders of their
total mass, extending Kahane's result for canonical cascades. Our main results
are analogues to the results by Kahane and Guivarc'h regarding the asymptotic
behavior of the right tail of the total mass. They rely on a new observation
made about the cones used to define the log-infinitely divisible cascades; this
observation provides a "non-independent" random difference equation satisfied
by the total mass of the measures. The non-independent structure brings new
difficulties to study the random difference equation, which we overcome thanks
to Goldie's implicit renewal theory. We also discuss the finiteness of moments
of negative orders, and some geometric properties of the support.
