Bochner’s theorem tells us each positive semidefinite function is the characteristic function of a probability measure. (see, for example, these lecture notes). In combination with it's converse, which follows from associated definitions, we can see that positive definite functions are in one to one correspondence with random variables' probability distributions.
All the resources I can find online treat only the univariate case, but I'm interested in joint distributions of random variables. Could someone help me understand how this result generalizes or fails to generalize to the multivariate case?