Limit theorems arising from sequences of multinomial random vectors

Given a sequence of i.i.d. multinomial random vectors, each of the coordinates of the sum of the random vectors, multiplied by their respective indices in the sequence, is centered and normed by its conditional mean and conditional standard deviation, conditioned on the sum of the random vectors. Multivariate normal and [chi]2 limit distributions are then obtained. Further, limit distributions are determined for sequences of sums of certain diagonal affine transformations of triangular arrays of multinomial random vectors. These distributions involve those of the multivariate normal and the compound Poisson. Necessary and sufficient conditions are obtained for convergence to the multivariate normal distribution.