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A Criterion of Boundedness of Discrete Branching Random Walk

  • F. I. Karpelevich
  • Y. M. Suhov
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)

Abstract

A general model of a branching random walk in Z is considered, where the branching and displacements occur with probabilities determined by the position of a parent. A necessary and sufficient condition is given for the random variable
$$ M = \sup \;\max \;\;{X_{n,k}}\;\;\;n \geqslant 0,1 \leqslant k \leqslant {N_n} $$
(1)
to be finite. Here X n,k is the position of the k-th offspring in the n-th generation. The condition is stated in terms of a naturally arising linear functional equation. A number of examples are discussed, where the condition may be verified.

Key words

branching random walk boundedness linear functional equation non-negative unbounded solutions excessive functions 

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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • F. I. Karpelevich
    • 1
  • Y. M. Suhov
    • 2
    • 3
  1. 1.Moscow Institute of Transport EngineeringThe Russian Ministry of Railway TransportMoscowRussia
  2. 2.Statistical Laboratory, Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeEngland, UK
  3. 3.St John’s CollegeCambridgeEngland, UK

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