  A statistical test of independence may be constructed using the
Hilbert-Schmidt Independence Criterion (HSIC) as a test statistic. The HSIC is
defined as the distance between the embedding of the joint distribution, and
the embedding of the product of the marginals, in a Reproducing Kernel Hilbert
Space (RKHS). It has previously been shown that when the kernel used in
defining the joint embedding is characteristic (that is, the embedding of the
joint distribution to the feature space is injective), then the HSIC-based test
is consistent. In particular, it is sufficient for the product of kernels on
the individual domains to be characteristic on the joint domain. In this note,
it is established via a result of Lyons (2013) that HSIC-based independence
tests are consistent when kernels on the marginals are characteristic on their
respective domains, even when the product of kernels is not characteristic on
the joint domain.
