On the existence of probability distributions with given marginals

Let X = {0,..., n - 1} and Gamma = {(x(1),..., x(s))} is an element of X-s : Sigma(sigma=1)(s) x(sigma) = n - 1. For the marginals of probability distributions on Gamma with the additional property of forming an s-tuple of decreasing probabilities on X a simple characterization is given. This has an interesting application to asymptotic spectra in the sense of Strassen (J. Reine Angew. Math., 384 (1988), pp. 102-152; 413 (1991), pp. 127-180]. Some correlated questions are discussed in an appendix.