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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
##### Keywords:
random walk; random scenery; large deviations; local times
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