A remark on the CLT for a random walk in a random environment
The author gives an elementary proof of an almost sure central limit theorem for a particular model of a \(d\)-dimensional random walk in random environment introduced earlier by \textit{C. Boldrighini, R. A. Minlos} and \textit{A. Pellegrinotti} [BMP] [Probab. Theory Relat. Fields 109, No. 2, 245--273 (1997; Zbl 0888.60061)]. The model is as follows. Let \(\alpha=(\alpha_ n(x))_{n\in\mathbb N_ 0,x\in \mathbb Z^ d}\) be a sequence of i.i.d. random variables with marginal distribution \(\nu\) defined on some measurable space \((S,\mathcal S)\). Furthermore, \(c:\mathbb Z^ d\times S\to\mathbb R\) is a measurable map satisfying \(\sum_x c(x,s)=0\) for any \(s\) and \(\int_ S c(x,s)\,\nu(ds)=0\) for any \(x\). Let \(p_0\) be a probability distribution on \(\mathbb Z^ d\) with mean vector \(b\in\mathbb R^ d\), finite second moments and such that the characteristic function of \(p_0\) is strictly smaller than one outside \(2\pi\mathbb Z^ d\). For consistency, assume that \(p_ 0(x)+c(x,s)\in[0,1]\) for any \(x\) and \(s\). The random walk in random environment, \((X_ n)_{n\in\mathbb N_ 0}\), is defined by \(X_ 0=0\) and the following rule. Conditional on the `environment\'\ \(\alpha\), \[ P^\alpha(X_{n+1}=y\mid X_ n=x)=p_ 0(y-x)+\varepsilon c(y-x,\alpha_ n(x)). \] For this model, [BMP] proved an almost sure central limit theorem: For \(\varepsilon>0\) sufficiently small, \((X_ n-nb)/\sqrt n\) converges in distribution under \(P^\alpha\), for almost all \(\alpha\), towards the normal distribution with the covariance matrix of the steps of the free walk. Under the assumption that the many-step distribution of the free random walk (i.e., of the one for \(\varepsilon=0\)) satisfies a Gaussian estimate, the author reproves this result, thereby simplifying the proof of [BMP] and using only elementary methods.
scientific article; zbMATH DE number 2137171