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Abstract

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.

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Let us consider a triangular array of random vectors (X (n) j; Y (n) j), n = 1;2;: : :, 1 5 j 5 kn, such that the first coordinates X (n) j take their values in a non-compact Lie group and the second coordinates Y (n) j in a compact group. Let the random vectors (X (n) j; Y (n) j) be independent for fixed n, but we do not assume any (independence type) condition about the relation between the components of these vectors. We show under fairly general conditions that if both random products Sn = kn Q j=1 X (n) j and Tn = kn Q j=1 Y (n) j have a limit distribution, then also the random vectors (Sn; Tn) converge in distribution as n !1 . Moreover, the non-compact and compact coordinates of a random vector with this limit distribution are independent.

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^ n , α . Then the assertion of Lemma 10 can be written in terms of convergence in distribution as (12) X N , n , α → N → ∞ D X ^ n , α . As it was mentioned above, the series for ζ n ( s,α ) is absolutely convergent for σ > 1 2 . Therefore, in this

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