On generalized multiplicative cascades

We consider a generalized Mandelbrotʹs martingale {Yn} and the associated Mandelbrotʹs measure μω on marked trees. If the limit variable Z=lim Yn is not degenerate, we study the asymptotic behavior at infinity of its distribution; in the contrary case, we prove that there is an associated natural martingale Yn∗ converging to a non-negative random variable Z∗ with infinite mean. Both Z and Z∗ lead to non-trivial solution of a distributional equation which extends the notion of stable laws. Precise results are obtained about Hausdorff measures and packing measures of the support of the Mandelbrotʹs measure.