Abstract
Let \(\{S_k:k\ge 0\}\) be a symmetric and aperiodic random walk on \(\mathbb {Z}^d\), \(d\ge 3\), and \(\{\xi (z),z\in \mathbb {Z}^d\}\) a collection of independent and identically distributed random variables. Consider a random walk in random scenery defined by \(T_n=\sum _{k=0}^n\xi (S_k)=\sum _{z\in \mathbb {Z}^d}l_n(z)\xi (z)\), where \(l_n(z)=\sum _{k=0}^nI{\{S_k=z\}}\) is the local time of the random walk at the site z. Using \((\sum _{z\in \mathbb {Z}^d}l_n(z)|\xi (z)|^p)^{1/p}\), \(p\ge 2\), as the normalizing constants, we establish self-normalized moderate deviations for random walk in random scenery under a much weaker condition than a finite moment-generating function of the scenery variables.
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Acknowledgements
Ofer Zeitouni thanks Amine Asselah for an enlightening discussion that led to the construction of Example 2.1. He thanks the Chinese University of Hong Kong for their hospitality during his visit there in February 2018. His work was partially supported by an Israel Science Foundation Grant. Q.M. Shao’s research is partially supported by the Hong Kong RGC GRF 14302515 and 14304917.
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Feng, X., Shao, QM. & Zeitouni, O. Self-normalized Moderate Deviations for Random Walk in Random Scenery. J Theor Probab 34, 103–124 (2021). https://doi.org/10.1007/s10959-019-00965-2
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DOI: https://doi.org/10.1007/s10959-019-00965-2