Distance multivariance: New dependence measures for random vectors. (arXiv:1711.07775v2 [math.PR] UPDATED)
We introduce two new measures for the dependence of $n \ge 2$ random
variables: distance multivariance and total distance multivariance. Both
measures are based on the weighted $L^2$-distance of quantities related to the
characteristic functions of the underlying random variables. These extend
distance covariance (introduced by Sz\'ekely, Rizzo and Bakirov) from pairs of
random variables to $n$-tuplets of random variables. We show that total
distance multivariance can be used to detect the independence of $n$ random
variables and has a simple finite-sample representation in terms of distance
matrices of the sample points, where distance is measured by a continuous
negative definite function. Under some mild moment conditions, this leads to a
test for independence of multiple random vectors which is consistent against
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