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Journal Articles Stochastic Processes and their Applications Year : 2020

The heavy range of randomly biased walks on trees

Pierre Andreoletti
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Roland Diel

Abstract

We focus on recurrent random walks in random environment (RWRE) on Galton-Watson trees. The range of these walks, that is the number of sites visited at some fixed time, has been studied in three different papers [AC18], [AdR17] and [dR16]. Here we study the heavy range: the number of edges visited at least α times for some integer α. The asymptotic behavior of this process when α is a power of the number of steps of the walk is given for all the recurrent cases. It turns out that this heavy range plays a crucial role in the rate of convergence of an estimator of the environment from a single trajectory of the RWRE.
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Dates and versions

hal-01849187 , version 1 (25-07-2018)
hal-01849187 , version 2 (05-11-2018)
hal-01849187 , version 3 (19-12-2018)

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Pierre Andreoletti, Roland Diel. The heavy range of randomly biased walks on trees. Stochastic Processes and their Applications, 2020, 130 (2), pp.962-999. ⟨10.1016/j.spa.2019.04.004⟩. ⟨hal-01849187v3⟩
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