We introduce a simple variation of Doeblin's condition, Condition D*, that assures the uniform ergodicity of a Markov chain.
It is also shown that for non-homogeneous chains our conditions are equivalent to Dobrushin's weak ergodic coefficient.
Convergence in Mallows distance is of particular interest when heavy-tailed distributions are considered. For 1≦α<2, it constitutes an alternative technique to derive central limit type theorems for non-Gaussian α-stable laws. In this note, we further explore the connection between Mallows distance and convergence in distribution. Conditions for their equivalence are presented.