On generalized multiplicative cascades

We consider a generalized Mandelbrot's martingale {Y-n} and the associated Mandelbrot's measure mu(alpha) on marked trees. If the limit variable Z = lim Y-n 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 Y-n* 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 Mandelbrot's measure. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: Primary: 60J80, 60G57, 28A78, 28A80; Secondary: 60G42.