  The Hilbert--Schmidt Independence Criterion (HSIC) is a popular measure of
the dependency between two random variables. The statistic dHSIC is an
extension of HSIC that can be used to test joint independence of $d$ random
variables. Such hypothesis testing for (joint) independence is often done using
a permutation test, which compares the observed data with randomly permuted
datasets. The main contribution of this work is proving that the power of such
independence tests converges to 1 as the sample size converges to infinity.
This answers a question that was asked in (Pfister, 2018) Additionally this
work proves correct type 1 error rate of HSIC and dHSIC permutation tests and
provides guidance on how to select the number of permutations one uses in
practice. While correct type 1 error rate was already proved in (Pfister,
2018), we provide a modified proof following (Berrett, 2019), which extends to
the case of non-continuous data. The number of permutations to use was studied
e.g. by (Marozzi, 2004) but not in the context of HSIC and with a slight
difference in the estimate of the $p$-value and for permutations rather than
vectors of permutations. While the last two points have limited novelty we
include these to give a complete overview of permutation testing in the context
of HSIC and dHSIC.
