Asymptotic properties and absolute continuity of laws stable by random weighted mean

We study properties of stable-like laws, which are solutions of the distributional equation Z=d∑i=1NAiZi, where (N,A1,A2,…) is a given random variable with values in {0,1,…}×[0,∞)×[0,∞)×…, and Z,Z1,Z2,… are identically distributed positive random variables, independent of each other and independent of (N,A1,A2,…). Examples of such laws contain the laws of the well-known limit random variables in: (a) the Galton–Watson process or general branching processes, (b) branching random walks, (c) multiplicative processes, and (d) smoothing processes. For any solution Z (with finite or infinite mean), we find asymptotic properties of the distribution function P(Z⩽x) and those of the characteristic function EeitZ; we prove that the distribution of Z is absolutely continuous on (0,∞), and that its support is the whole half-line [0,∞). Solutions which are not necessarily positive are also considered.