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Self-normalized moderate deviations for random walk in random scenery. (English) Zbl 1469.60086

Summary: 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.

MSC:

60F10 Large deviations
60G50 Sums of independent random variables; random walks
60K37 Processes in random environments
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