Fixed points of the smoothing transform: The boundary case

Let A=(A₁,A₂,A₃,…) be a random sequence of non-negative numbers that are ultimately zero with E[∑A_i]=1 and E[∑A_ilogA_i]≤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[∏_iΦ(ψA_i)], where Φ is the Laplace transform of a non-negative random variable. The study complements, and extends, existing results on the case when E[∑A_ilogA_i]<0. New results on the asymptotic behaviour of the solutions near zero in the boundary case, where E[∑A_ilogA_i]=0, are obtained.