Let \(X_ n\) have a k-dimensional binomial distribution \(B_ k(n,p_ n)\), \(p_ n=(p_{nl},...,p_{nk})\), the distribution of the sum of n independent and identically distributed k-dimensional random vectors that take all their values in the set of vertices of the unit cube in \(R^ k\) (not necessarily just those on the axes, that is the multinomial special case), and let X have a k-dimensional distribution not supported by any lower-dimensional hyperplane. In this high-quality paper the authors prove the following results.
(i) If \(X_ n\) converges in distribution to X (as \(n\to \infty)\), then the distribution of X is k-variate Poisson.
(ii) If diagonal affine transforms of \(X_ n\) converge in distribution to X, then X has the distribution of a diagonal affine transform of a Poisson-Gaussian random vector.
(iii) If general affine transforms of \(X_ n\) converge in distribution to X and \(X_ n\) satisfies a ”non-collapsibility” condition, then X has the distribution of an invertible affine transform of a Poisson-Gaussian vector.
(iv) The conclusion of (iii) is not true in general without non- collapsibility.
(V) If all the k sequences \(np_{ni}\) are bounded, then \(X_ n\) is necessarily non-collapsible in (iii).