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

Abstract

We study properties of stable-like laws, which are solutions of the distributional equation Z = d ∑ i=1 NA iZ i, where ( N, A 1, A 2,…) is a given random variable with values in {0,1,…}×[0,∞)×[0,∞)×…, and Z, Z 1, Z 2,… are identically distributed positive random variables, independent of each other and independent of ( N, A 1, A 2,…). 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 Ee i tZ ; 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.