Let A = (A1,A2, A3,...) be a random sequence of non-negative numbers that are ultimately zero with E[? A i] = 1 and E[?Ai log Ai= 0. The uniqueness of the non-negative fixed points of the associated smoothing transform is considered. These fixed points are solutions to the functional equation f(?) = E[?if(?Ai)], where f is the Laplace transform of a non-negative random variable. The study complements, and extends, existing results on the case when E[?A ilog Ai 0. New results on the asymptotic behaviour of the solutions near zero in the boundary case, where E[?Ailog A i 0, are obtained.
|Journal||Electronic Journal of Probability|
|Publication status||Published - 2005|
- Branching random walk
- Functional equation
- Smoothing transform