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Random Walks on Trees with Finitely Many Cone Types

Author

Listed:
  • Tatiana Nagnibeda

    (Royal Institute of Technology)

  • Wolfgang Woess

    (Technische Universität Graz)

Abstract

This paper is devoted to the study of random walks on infinite trees with finitely many cone types (also called periodic trees). We consider nearest neighbour random walks with probabilities adapted to the cone structure of the tree, which include in particular the well studied classes of simple and homesick random walks. We give a simple criterion for transience or recurrence of the random walk and prove that the spectral radius is equal to 1 if and only if the random walk is recurrent. Furthermore, we study the asymptotic behaviour of return probabilitites and prove a local limit theorem. In the transient case, we also prove a law of large numbers and compute the rate of escape of the random walk to infinity, as well as prove a central limit theorem. Finally, we describe the structure of the boundary process and explain its connection with the random walk.

Suggested Citation

  • Tatiana Nagnibeda & Wolfgang Woess, 2002. "Random Walks on Trees with Finitely Many Cone Types," Journal of Theoretical Probability, Springer, vol. 15(2), pages 383-422, April.
    • Handle: RePEc:spr:jotpro:v:15:y:2002:i:2:d:10.1023_a:1014810827031
    DOI: 10.1023/A:1014810827031
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    Citations

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    Cited by:

    1. Lorenz A. Gilch & Sebastian Müller, 2011. "Random Walks on Directed Covers of Graphs," Journal of Theoretical Probability, Springer, vol. 24(1), pages 118-149, March.
    2. Michael Björklund, 2010. "Central Limit Theorems for Gromov Hyperbolic Groups," Journal of Theoretical Probability, Springer, vol. 23(3), pages 871-887, September.

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