Couplings with prescribed maxima: A short proof of the quasi-copula characterization

We provide a short probabilistic proof of the classical characterization of multivariate quasi-copulas, addressing the call for a simpler and more transparent proof expressed by Sempi (2017). Our approach is based on a coupling construction for random variables with fixed marginal distributions and a prescribed distribution of their maximum. In our main result, we show that such a coupling exists if and only if the maximum is stochastically no smaller than each component and the distribution of the maximum is dominated by the sum of the marginal measures. The construction applies to arbitrary distributions and relies only on a decomposition of the maximum law and monotone quantile couplings. A direct consequence of our main result is the existence of copulas with prescribed values along an increasing track.