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Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities

  • A. M. Iksanov
  • U. Rösler
Article

Abstract

Let \(\mathcal{M}^{(n)} \), n = 1, 2, ..., be the supercritical branching random walk in which the family sizes may be infinite with positive probability. Assume that a natural martingale related to \(\mathcal{M}^{(n)} \) converges almost surely and in the mean to a random variable W. For a large subclass of nonnegative and concave functions ƒ, we provide a criterion for the finiteness of \(\mathbb{E}\) Wf(W). The main assertions of the present paper generalize some results obtained recently in Kuhlbusch’s Ph.D. thesis as well as previously known results for the Galton-Watson processes. In the process of the proof, we study the existence of the ƒ-moments of perpetuities.

Keywords

Random Walk Concave Function Borel Function Independent Copy Moment Result 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. M. Iksanov
    • 1
  • U. Rösler
    • 2
  1. 1.Shevchenko Kiev National UniversityKiev
  2. 2.University of KielKielGermany

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