Conditions for weak ergodicity of inhomogeneous Markov chains

A classical result of Wolfowitz states that an inhomogeneous Markov chain is weakly ergodic if the transition matrices are drawn from a finite set of indecomposable and aperiodic matrices and the products of transition matrices are also indecomposable and aperiodic. Since products of indecomposable and aperiodic matrices can be decomposable, any finite set of indecomposable and aperiodic transition matrices does not guarantee weak ergodicity. We present conditions for weak ergodicity which are simpler to verify and are related to properties of the graph of the transition matrices.