Fixed Points of the Smoothing Transform: the Boundary Case

Let $A=(A_1,A_2,A_3,\ldots)$ be a random sequence of non-negative numbers that are ultimately zero with $E[\sum A_i]=1$ and $E \left[\sum A_{i} \log A_i \right] \leq 0$. The uniqueness of the non-negative fixed points of the associated smoothing transform is considered. These fixed points are solutions to the functional equation $\Phi(\psi)= E \left[ \prod_{i} \Phi(\psi A_i) \right], $ where $\Phi$ is the Laplace transform of a non-negative random variable. The study complements, and extends, existing results on the case when $E\left[\sum A_{i} \log A_i \right]<0$. New results on the asymptotic behaviour of the solutions near zero in the boundary case, where $E\left[\sum A_{i} \log A_i \right]=0$, are obtained.