Fixed points of the smoothing transform: the boundary case. (English) Zbl 1110.60081

Biggins, J. D.; Kyprianou, A. E.

Electron. J. Probab. 10, Paper No. 17, 609-631 (2005).

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