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A note on random walk in random scenery. (English) Zbl 1112.60088
Summary: We consider a random walk in random scenery \(\{X_n= \eta(S_0+\cdots+ \eta(S_n)\), \(n\in\mathbb N\}\), where a centered walk \(\{S_n\), \(n\in\mathbb N\}\) is independent of the scenery \(\{\eta(x)\), \(x\in\mathbb Z^d\}\), consisting of symmetric i.i.d. with tail distribution \(P(\eta(x)>t)\sim \exp(-c_\alpha t^\alpha)\), with \(\leq\alpha< d/2\). We study the probability, when averaged over both randomness, that \(\{X_n>ny\}\) for \(y>0\), and \(n\) large. We show that the large deviation estimate is of order \(\exp(-c(ny)^a)\), with \(a=\alpha/(\alpha+1)\).

MSC:
60K37 Processes in random environments
60F10 Large deviations
60J55 Local time and additive functionals
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