论文标题
中央限制定理用于光滑紧凑的Riemannian歧管中的内在特征平均值
Central limit theorem for intrinsic Frechet means in smooth compact Riemannian manifolds
论文作者
论文摘要
我们证明了在紧凑的riemannian歧管中的独立和相同分布观测值的特征平均值的中心限制定理（CLT），假设人口特里切特的平均值是独特的。以前的一般CLT结果在这种情况下，假设Frechet的切割基因均位于人口分布的支持之外。据我们所知，本文中的CLT是第一篇允许切割轨迹在分布的支持中具有1或两个的限量。证明的一个关键部分是建立一个渐近近似，以平行某个向量场的平行运输。 CLT中是否出现非标准术语取决于剪切基因座的共数是否是一个或大于一个：在前一种情况下，出现了非标准术语，但在后一种情况下却没有。这是第一篇为非标准项提供一般和明确表达式的纸张，该术语是当切割基因座的共数是一个时出现的。
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.
