Weak convergence of joint distributions generally does not imply convergence of conditional distributions. In particular, conditional distributions need not converge when joint Gaussian distributions converge to a singular Gaussian limit. Algebraically, this is due to the fact that at singular covariance matrices, Schur complements are not continuous functions of the matrix entries. Our results lay out special conditions under which convergence of Gaussian conditional distributions nevertheless occurs, and we exemplify how this allows one to reason about conditional independence in a new class of graphical models.

翻译：联合分布的弱收敛通常不意味着条件分布的收敛。特别地，当联合高斯分布收敛于一个奇异高斯极限时，条件分布未必收敛。从代数角度看，这是因为在奇异协方差矩阵处，舒尔补并非矩阵元素的连续函数。我们的研究结果提出了高斯条件分布仍能收敛的特殊条件，并通过示例说明了这如何使得我们能够在一类新的图模型中推理条件独立性。