After reviewing the asymptotic variance stabilizing transformations in one dimension, a generalization of these to multivariate cases is discussed. Results are given for the uniqueness of solutions when they exist, but unlike the one-dimensional case, covariance stabilizing transformations need not exist. In the two-dimensional case, a necessary and sufficient condition is given for the existence of solutions. It takes the form of a second-order partial differential equation that the elements of any square root of the inverse of the limiting covariance matrix must satisfy. This condition is applied to three examples with the conclusion that no covariance stabilizing transformation exists for the trinomial distribution. It is conjectured that this non-existence of solutions is true for the general multinomial.
"Covariance Stabilizing Transformations." Ann. Statist. 1 (1) 84 - 92, January, 1973. https://doi.org/10.1214/aos/1193342384