We prove a central limit theorem (CLT) for the Frechet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Frechet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Frechet mean lies outside the support of the population distribution. So far as we are aware, the CLT in the present paper is the first which allows the cut locus to have co-dimension one or two when it is included in the support of the distribution. A key part of the proof is establishing an asymptotic approximation for the parallel transport of a certain vector field. Whether or not a non-standard term arises in the CLT depends on whether the co-dimension of the cut locus is one or greater than one: in the former case a non-standard term appears but not in the latter case. This is the first paper to give a general and explicit expression for the non-standard term which arises when the co-dimension of the cut locus is one.
翻译:我们用紧凑的Riemannian 方块中独立和同样分布的观测假设,Frechet 方块是独一无二的。以前通用的CLT在这一背景下的结果假设Frechet 的切角是人口分布所不能支持的。据我们所知,本文件中的CLT是第一个允许切角在支持分布时共分一或两个。证据的一个关键部分是确定某一矢量场平行运输的无症状近似值。CLT中是否出现非标准词取决于切角的共分界是否是一个或一个以上:在前一种情况中,出现非标准词,但在后一种情况中则不出现。这是第一个对切角共分点是一或两个时出现的非标准词作一般性和明确表达的文件。