An application of a multivariate central limit theorem to sampling without replacement

A general multivariate central limit theorem of M. G. Hahn, P. Hahn, and M. J. Klass (1983, Ann. Probab. 11 277-301) gives conditions relating the one-dimensional marginals of the summands in an infinitesimal triangular array to those of the limit law. This theorem is applied here to the sample sums obtained by sampling without replacement from a sequence of finite multivariate populations. An important subcase of this result is where the sequence of populations' values are restricted to a finite set. Here the limit laws are shown to be a natural generalization of the multivariate Poisson distribution.