Asymptotics of a dynamic random walk in a random scenery. I: Law of large numbers
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.
scientific article; zbMATH DE number 1453268