On some laws invariant by random weighted mean

We consider solutions of the the distributional equation (E): Z (d) double under bar Sigma(i=1)(N) A(i)Z(i), where (N, A(1), A(2),...) is a random var iable with values in {0, 1,...} x [0, infinity[x[0, infinity[x... (and with arbitrary joint law), and Z, Z(1),Z(2),... are positive random variables, independent each other and independent of (N, Al, Ap,...). Examples are the distributions of the famous limit random variables of the following proces ses: (a) the Bellman-Harris process [1,10, 19] and the Crump-Mode-Jagers pr ocess [11,9], Co) the branching mndorn walks [7,8], (c) the multiplicative cascades [21,17,14,20], (d) the smoothing processes [13]. For any solution Z (with finite or infinite mean), we find asymptotic properties of the dist ribution function P(Z less than or equal to x) at 0 and the characteristic function Ee(itZ) at infinity we prove that the distribution of Z is absolut e continuous on (0, infinity), and that its support is the whole half line [0,infinity). We therefore obtain new results for all processes mentioned a bove. (C 2000 Academie des sciences/Editions scientifiques et medicales Els evier SAS.