Hilbert space embeddings of independence tests of several variables

In this paper, we present the general theory of embedding independence tests on Hilbert spaces that generalizes the concepts of distance covariance, distance multivariance and HSIC. This is done by defining new types of kernel on an $n$ Cartesian product called positive definite independent of order $k$. An emphasis is given on the continuous case in order to obtain a version of the Kernel Mean Embedding for this new classes of kernels. We also provide $2$ explicit methods to construct examples for this new type of kernel on a general space by using Bernstein functions of several variables and completely monotone functions of higher order.