It is known that when a multivariate probability distribution has a “big” conditional quantile that is fully reproducible when narrowed to uni-variate quantiles, then the respective quantile differential equation is completely integrable. Yet the converse is not true in general. In this paper, we show that, provided that certain conditions are satisfied, from the given completely integrable quantile equation, one can construct a multivariate distribution with a fully reproducible “big” conditional quantile. The constructed probability distribution may differ from the original distribution. We will also give a generalization of this result that suggests a method of shifting from an \((n-1)\)-dimensional distribution satisfying certain conditions to an n-dimensional distribution with a fully reproducible “big” conditional quantile.

众所周知，当多元概率分布具有“大”条件分位数时，当缩小到单变量分位数时可以完全重现时，则各个分位数微分方程是完全可积分的。然而，通常情况并非如此。在本文中，我们表明，只要满足某些条件，就可以从给定的完全可分位数方程，构造具有完全可重现的“大”条件分位数的多元分布。构造的概率分布可能与原始分布不同。我们还将对此结果进行概括，提出一种将满足特定条件的\（（（n-1）\）-维分布转移到n具有完全可重现的“大”条件分位数的三维分布。