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Asymptotics of a dynamic random walk in a random scenery. I: Law of large numbers. (English) Zbl 0969.60045
Define a non-homogeneous Markov chain $$(S_n)_{n\in\mathbb N_0}$$ on $$\mathbb Z$$ as follows. Fix some $$x$$ in the $$d$$-dimensional torus $$\mathbb T^d$$ and a vector $$\alpha\in \mathbb R^d$$ having irrational components and some function $$f:\mathbb T^d\to [0,1]$$. Put $$S_0=0$$, and the steps $$(S_i-S_{i-1})$$ of the walk are assumed to be independent and to assume the value 1 with probability $$f({\tau_\alpha}^ix)$$ and the value $$-1$$ otherwise, where $$\tau_\alpha: {\mathbb T}^d\to\mathbb T^d$$ is the rotation by $$\alpha$$ on the torus. Furthermore, let $$(\xi(z))_{z\in\mathbb Z}$$ be an i.i.d. sequence of real random variables, acting as a random scenery, and define a random walk in random scenery by $$Z_n=\sum_{i=0}^n\xi(S_i)$$. The main goal of the paper is the proof for the facts that $$(S_n)_{n}$$ is recurrent on its moving average (i.e., $$P(\limsup_{n\to\infty} \{|S_n-ES_n|<\varepsilon\})=1$$ for any $$\varepsilon>0$$), and that $$(Z_n)_{n}$$ satisfies a weak law of large numbers.
The results are formulated more precisely as follows. Let $$f$$ be of bounded variation in the sense of Hardy and Krause, and assume that $$a=4\int_{\mathbb T^d}f(t)(1-f(t)) dt$$ is positive and that $$\int_{\mathbb T^d}f(t) dt=1/2$$. (These two conditions ensure, via an ergodic theorem, that the limiting drift of the walk is zero and that the limiting variance is positive.) Then $$(S_n)_{n}$$ is recurrent on its moving average, and (under some technical additional assumption) $$P(S_{2n}=0)\sim (a\pi n)^{-1/2}$$ and $$Z_n/n\to 0$$ in probability. Furthermore, in the case that the components of $$\alpha$$ are rational, some natural sufficient conditions are given such that the recurrence of $$(S_n)_n$$ can be decided and such that $$Z_n/n\to 0$$ almost surely.
Reviewer: W.König (Berlin)

##### MSC:
 60G50 Sums of independent random variables; random walks
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