Conditions for equivalence between Mallows distance and convergence to stable laws

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.