I'm trying to bound the KL divergence between two continuous random variables with the KL divergence between two Gaussian mixture approximations motivated by the fact that the Gaussian mixture model is a universal approximator of densities, see https://stats.stackexchange.com/questions/365155/a-gaussian-mixture-model-is-a-universal-approximator-of-densities/365158#365158
In the end I would like to plug in the KL divergence for Gaussians.
So my question: Are there any results about the parameters of the mixture model approximating a given density and how the KL divergence between a function and its Gaussian mixture approximation behaves?