On the existence of probability distributions with given marginals

Let $X=\{0,\ldots, n-1\}$ and $\Gamma=\{(x_1,\ldots, x_s)\}\in X^s\colon\,\sum_{\sigma=1}^s x_\sigma=n-1$. 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.