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 = l im Y-n is not degenerate, we study the asymptotic behavior at infinity of i ts distribution; in the contrary case, we prove that there is an associated natural martingale Y-n* converging to a non-negative random variable Z* wi th infinite mean. Both Z and Z* lead to non-trivial solution of a distribut ional equation which extends the notion of stable laws. Precise results are obtained about Hausdorff measures and packing measures of the support of t he Mandelbrot's measure. (C) 2000 Elsevier Science B.V. All rights reserved . MSC: Primary: 60J80, 60G57, 28A78, 28A80; Secondary: 60G42.