Fixed points of the smoothing transform: The boundary case

J. D. Biggins, A. E. Kyprianou

Research output: Contribution to journalArticlepeer-review

53 Citations (Scopus)


Let A = (A1,A2, A3,...) be a random sequence of non-negative numbers that are ultimately zero with E[? A i] = 1 and E[?Ai log Ai= 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 f(?) = E[?if(?Ai)], where f is the Laplace transform of a non-negative random variable. The study complements, and extends, existing results on the case when E[?A ilog Ai 0. New results on the asymptotic behaviour of the solutions near zero in the boundary case, where E[?Ailog A i 0, are obtained.

Original languageEnglish
JournalElectronic Journal of Probability
Publication statusPublished - 2005


  • Branching random walk
  • Functional equation
  • Smoothing transform


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