Detecting independence of random vectors II. Distance multivariance and Gaussian multivariance
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We introduce two new measures for the dependence of n > 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. They extend distance covariance (introduced by Szekely, Rizzo and Bakirov) and generalized distance covariance (introduced in part I) 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. Based on our theoretical results, we present a test for independence of multiple random vectors which is consistent against all alternatives.
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