Stein's method has been widely used to achieve distributional approximations for probability distributions defined in Euclidean spaces. Recently, techniques to extend Stein's method to manifold-valued random variables with distributions defined on the respective manifolds have been reported. However, several of these methods impose strong regularity conditions on the distributions as well as the manifolds and/or consider very special cases. In this paper, we present a novel framework for Stein's method on Riemannian manifolds using the Friedrichs extension technique applied to self-adjoint unbounded operators. This framework is applicable to a variety of conventional and unconventional situations, including but not limited to, intrinsically defined non-smooth distributions, truncated distributions on Riemannian manifolds, distributions on incomplete Riemannian manifolds, etc. Moreover, the stronger the regularity conditions imposed on the manifolds or target distributions, the stronger will be the characterization ability of our novel Stein pair, which facilitates the application of Stein's method to problem domains hitherto uncharted. We present several (non-numeric) examples illustrating the applicability of the presented theory.
翻译:施泰因的方法已被广泛用于实现欧几里德空间内所定义的概率分布分布分布分布的分布近似值。 最近,报告了将施泰因的方法推广到在不同的方块上所定义分布的多值随机变数的技术。 但是,其中一些方法对分布以及元件和/或非常特殊的情况规定了严格的常规性条件。 在本文中,我们提出了一个用于施泰因方法的新型框架,用于使用Friedrichs扩展技术,用于不受约束的自联操作者。这个框架适用于各种常规和非常规情况,包括但不限于:内在界定的非mooth分布、Riemannian元件上的短径分布、不完全的里曼元的分布等等。此外,对管道或目标分布所强加的常规性条件越强,我们新型施泰因配方的定性能力就越强,这有利于施泰因方法在迄今尚未描述的问题领域的应用。我们提出了几个(非数字性)例子,说明所提出的理论的适用性。