An extension of Kesten's renewal theorem for random walk in a random environment (Q1081967)

Let \(X_ 0,X_ 1,..\). be a random walk on \({\mathbb{Z}}\) with \(P(X_{n+1}=i+1| X_ n=i)=\alpha_ i\), \(P(X_{n+1}=i-1| X_ n=i)=1-\alpha_ i\), where \(\alpha_ n\), \(n\in {\mathbb{Z}}\), are the values of an i.i.d. sequence of random variables. Let \(\alpha\) (k) denote the sequence \(\alpha\) shifted by k, \(\alpha (k)_ n=\alpha_{k+n}\). The title refers to a weak convergence result on \(\alpha (X_ n)\); using a coupling device the paper deals with convergence in probability and almost sure convergence of the random variates \(E[f(\alpha (X_ n))| \alpha]\) for continuous functions f: [0,1]\({}^{{\mathbb{Z}}}\to {\mathbb{R}}\).