We prove a version of the multidimensional Fourth Moment Theorem for chaotic random vectors, in the general context of diffusion Markov generators. In addition to the usual componentwise convergence and unlike the infinite-dimensional Ornstein-Uhlenbeck generator case, another moment-type condition is required to imply joint convergence of of a given sequence of vectors.
"Multivariate Gaussian approximations on Markov chaoses." Electron. Commun. Probab. 21 1 - 9, 2016. https://doi.org/10.1214/16-ECP4615