Empirical distributions in a range of fields are often substantially non-Gaussian but smooth enough to suggest that the underlying population distribution has at least several derivatives. Linear combinations of many random variables often have smooth densities even if the random variables are not independent. The main result here generalizes the elementary result that a convolution of densities has as many derivatives as the sum of the number of derivatives of each component to certain nonlinear functions of random vectors whose components may not be independent. Linearity is replaced by an assumption that the nonlinear function has positive first partial derivatives bounded away from 0 along at least some coordinates and independence is replaced by an assumption that the joint density has certain mixed partial derivatives. This approach to justifying the smoothness of empirical distributions is contrasted with other possible approaches, including solutions of stochastic partial differential equations and ergodic distributions for chaotic dynamical systems.

一系列领域中的经验分布通常基本上是非高斯分布，但足够平滑以表明潜在的总体分布至少有几个导数。即使随机变量不是独立的，许多随机变量的线性组合通常也具有平滑的密度。这里的主要结果概括了基本结果，即密度卷积具有与每个分量的导数数量之和一样多的导数到随机向量的某些非线性函数，这些函数的分量可能不是独立的。线性被假设替代，即非线性函数具有沿至少一些坐标远离 0 的正一阶偏导数，而独立性被假设替代，即联合密度具有某些混合偏导数。