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Fixed points of the smoothing transform: the boundary case. (English) Zbl 1110.60081
Summary: Let $$A=(A_1,A_2,A_3,\ldots)$$ be a random sequence of nonnegative 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 nonnegative 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.

##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60G42 Martingales with discrete parameter
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