On the Lin condition in strong ratio limit theorems (Q1889678)

Let \((X_n,n\geq 0)\) be a Markov chain with state space \((E,{\mathcal B})\) and \(P\) be a transition operator. It is proved that for a wide class of Markov chains and, in particular, for many random walks on groups the formula (Lin condition) \(\liminf (v(P^{n+1}f)/ v(P^nf))=1\) holds, where \(v\) and \(f\) are a probability measure on \({\mathcal B}\) and a \({\mathcal B}\)-measurable function, respectively.