Let $S_{p,n}$ denote the sample covariance matrix based on $n$ independent identically distributed $p$-dimensional random vectors in the null-case. The main result of this paper is an expansion of trace moments and power-trace covariances of $S_{p,n}$ simultaneously for both high- and low-dimensional data. To this end we develop a graph theory oriented ansatz of describing trace moments as weighted sums over colored graphs. Specifically, explicit formulas for the highest order coefficients in the expansion are deduced by restricting attention to graphs with either no or one cycle. The novelty is a color-preserving decomposition of graphs into a tree-structure and their seed graphs, which allows for the identification of Euler circuits from graphs with the same tree-structure but different seed graphs. This approach may also be used to approximate the mean and covariance to even higher degrees of accuracy.

中文翻译：

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带有图形着色的样本协方差矩阵的迹矩

令$S_{p,n}$ 表示在零情况下基于$n$ 独立同分布$p$ 维随机向量的样本协方差矩阵。本文的主要结果是同时扩展了高维和低维数据的$S_{p,n}$的迹矩和功率迹协方差。为此，我们开发了一种面向图论的 ansatz，将轨迹矩描述为彩色图上的加权和。具体来说，通过将注意力限制在没有循环或只有一个循环的图上，推导出了展开中最高阶系数的显式公式。新颖性是将图保存颜色分解为树结构及其种子图，这允许从具有相同树结构但不同种子图的图识别欧拉电路。