  In this paper, we study sample size thresholds for maximum likelihood
estimation for tensor normal models. Given the model parameters and the number
of samples, we determine whether, almost surely, (1) the likelihood function is
bounded from above, (2) maximum likelihood estimates (MLEs) exist, and (3) MLEs
exist uniquely. We obtain a complete answer for both real and complex models.
One consequence of our results is that almost sure boundedness of the
log-likelihood function guarantees almost sure existence of an MLE. Our
techniques are based on invariant theory and castling transforms.
