Limit theorem for a random walk in a reversible random environment
Let (\(\Omega\),\({\mathcal F},\mu)\) be a probability space and \(P^{(\omega)}\) be some probability measure associated with a homogeneous Markov chain X(k), \(k=0,1,...\), with values in \(Z^ d\) (d\(\geq 1)\) and with random transition probabilities. Denote by \(E^{\omega}(\cdot)\) and \(<\cdot >\) the means under the measures \(P^{(\omega)}\) and \(\mu\), respectively. Under some conditions on ``reversibility'' and ``small fluctuations'' the convergence of the process \(X_{\alpha,N}=X(N\cdot)/N^{1/\alpha}\) in the \(\mu\)-mean to a stable process \(W_{\alpha}(\cdot)\) with index \(\alpha\in (1,2)\) is obtained, i.e. for any finite set of points \(\{t_ i\}\subset (0,\infty)\) and any continuous, finite function f: \[ <| E^{(\omega)}f(X_{\alpha,N}(t_ 1),...,X_{\alpha,N}(t_ n))-E f(W_{\alpha}(t_ 1),...,W_{\alpha}(t_ n))| >\to 0\text{ for } N\to \infty. \]
scientific article; zbMATH DE number 4169795
scientific article