Correction: Discussion of Brownian distance covariance
Michael R. Kosorok · The Annals of Applied Statistics · 2013
The proof of Lemma 3 in Kosorok ( 2009) is incorrect since the second equality in the display of the proof is, in fact, an inequality (≤).Some results in Dueck et al. ( 2012) also indicate that Lemma 3 is not true.Moreover, it is not hard to obtain simple counterexamples.For example, if X ( 2) is a Rademacher random variable [i.e., P (X (2) = -1) = P (X (2) = 1) = 1/2] and X ( 3) is zero with probability 1, then f X(t ) = cos(t) and the inequality is strict for all t for which | cos(t)| = 1.Nevertheless, the conclusions of Lemma 6 in Kosorok ( 2009) remain valid, as shown in Lyons ( 2013), under even weaker conditions than those given in the statement of the lemma.Moreover, the other results of Kosorok ( 2009) are unaffected by Lemma 3 and thus remain valid.