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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.