A generalized Isserlis theorem for location mixtures of Gaussian random vectors

In a recent paper, Michalowicz et al. provide an extension of Isserlis theorem to the case of a Bernoulli location mixture of a Gaussian vector. We extend here this result to the case of any location mixture of Gaussian vector; we also provide an example of the Isserlis theorem for a “scale location” mixture of Gaussian, namely the d-dimensional generalized hyperbolic distribution.