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2021 Moderate deviations for the self-normalized random walk in random scenery
Tal Peretz
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Electron. J. Probab. 26: 1-16 (2021). DOI: 10.1214/21-EJP607


Let G be an infinite connected graph with vertex set V. Let {Sn:nN0} be the simple random walk on G and let {ξ(v):vV} be a collection of i.i.d. random variables which are independent of the random walk. Define the random walk in random scenery as Tn=k=0nξ(Sk), and the normalization variables Vn=(k=0nξ2(Sk))12 and Ln,2=(vVn2(v))12. For G=Zd and G=Td, the d-ary tree, we provide large deviations results for the self-normalized process Tnn(Ln,2Vn) under only finite moment assumptions on the scenery.

Funding Statement

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 692452).


I would like to thank my M.Sc. advisor Ofer Zeitouni for his guidance and support throughout all stages of this work. Thank you to both reviewers for providing detailed feedback that greatly improved the content and flow of the manuscript.


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Tal Peretz. "Moderate deviations for the self-normalized random walk in random scenery." Electron. J. Probab. 26 1 - 16, 2021.


Received: 9 January 2020; Accepted: 12 March 2021; Published: 2021
First available in Project Euclid: 7 April 2021

arXiv: 2001.05736
Digital Object Identifier: 10.1214/21-EJP607

Primary: 60F10
Secondary: 60G50 , 60K37

Keywords: Local times , Moderate deviations , Random walk in random scenery , self-normalized partial sums


Vol.26 • 2021
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