The smoothing transform; the boundary case

Abstract

Let A = (A1, A2, A3, . . .) be a random decreasing sequence of non-negative numbers that are ultimately zero with E[E Ai] = 1 and E [ E Ai log Ai] = 0. The uniqueness and properties of the non-negative fixed points of the associated smoothing transform are considered. These are solutions to the functional equation ø ( v ) = E [ I I i ø(vAi)] , where ø is the Laplace transform of a nonnegative random variable. The study complements existing results on the case when E [E Ai log Ai]< 0.