On Weak and Strong Ergodicity

We establish some classical results on ergodicity of inhomogeneous Markov chains in a more general context. Madsen (Ann Math Stat 42(1):405-408, 1971) extended the characterization of weak ergodicity presented by Paz (Ann Math Stat 41(2):539-550, 1970) from the countable case to the continuous case, admitting, however, that the transition probability functions have associated density functions. Here, we show that Madsen's extension remains valid if the transition probability functions are not associated with density functions. In addition, we extend the criteria for strong ergodicity of Dorea and Rojas Cruz (Sankhya Indian J Stat (2003-2007) 66(2):243-252, 2004) which is a refinement of a classical result given by Isaacson and Madsen (Markov chains, theory and applications, Wiley, New York, 1976).