Fixed points of the smoothing transform: The boundary case

Abstract

Let A = (A1, A2, A3,...) be a random sequence of non-negative numbers that are ultimately zero with E[∑ Ai] = 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 Φ(ψ) = E [Πi Φ(Π Ai)], where Φ is the Laplace transform of a non-negative random variable. The study complements, and extends, existing results on the case when E[∑ Ai log Ai] < 0. New results on the asymptotic behaviour of the solutions near zero in the boundary case, where E[∑ Ai log Ai] = 0, are obtained. © 2005 Applied Probability Trust.