 Favorite sites of randomly biased walks on a supercritical Galton–Watson treeDayue Chen, Loïc de Raphélis and Yueyun Hu Stochastic Processes and their Applications, 2018, vol. 128, issue 5, 1525-1557 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. Keywords: Biased random walk on the Galton–Watson tree; Local times; Favorite sites; Multitype Galton–Watson tree (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:128:y:2018:i:5:p:1525-1557 Ordering information: This journal article can be ordered from DOI: 10.1016/j.spa.2017.08.002 Access Statistics for this article Stochastic Processes and their Applications is currently edited by T. Mikosch More articles in Stochastic Processes and their Applications from Elsevier |
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