An extension of a functional equation of Poincaré and Mandelbrot

We consider the distributional equation Z -A\ + ... 4-A^Z^, where iV, Ai, Z, Zi are independent real random variables with N 6 N and Ai > 0. Each Ai is equally distributed as Ai and each Zi as Z; the distribution of Z is unknown. After establishing theorems about existence and uniqueness of solution, we show asymptotic properties at 0 of the distribution function P[Z < x], and those at 00 of the characteristic function Ee ltz ; such properties are also found for a random difference equation. In particular, we prove the absolute continuity of the distribution of Z under simple moment conditions. As applications, new results are established for multiplicative cascades, for age-dependent branching processes and for branching random walks.