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Favorite sites of randomly biased walks on a supercritical Galton–Watson tree

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  • Chen, Dayue
  • de Raphélis, Loïc
  • Hu, Yueyun

Abstract

Erdős and Révész (1984) initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Galton–Watson tree. We prove that there is some parameter κ∈(1,∞] such that the set of the favorite sites of the biased walk is almost surely bounded in the case κ∈(2,∞], tight in the case κ=2, and oscillates between a neighborhood of the root and the boundary of the range in the case κ∈(1,2). Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case κ∈(2,∞]. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton–Watson trees.

Suggested Citation

  • Chen, Dayue & de Raphélis, Loïc & Hu, Yueyun, 2018. "Favorite sites of randomly biased walks on a supercritical Galton–Watson tree," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1525-1557.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:5:p:1525-1557
    DOI: 10.1016/j.spa.2017.08.002
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    References listed on IDEAS

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    1. Afanasyev, V. I., 2001. "On the maximum of a subcritical branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 93(1), pages 87-107, May.
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    3. Eisenbaum, Nathalie & Khoshnevisan, Davar, 2002. "On the most visited sites of symmetric Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 241-256, October.
    4. Jagers, Peter, 1989. "General branching processes as Markov fields," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 183-212, August.
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