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.

翻译：Let $S ⁇ p, n} $ 表示基于美元独立分布的样本共变矩阵, 以美元为单位。 本文的主要结果就是将微量时间和能量- 振幅共变数同时扩展为 $S ⁇ p, n} 美元, 用于高维和低维数据。 为此, 我们开发了一个图形理论, 将微量描述为彩色图形的加权总和 。 具体地说, 扩展中最高顺序系数的清晰公式通过限制对无循环或一个循环的图形的注意来推导。 新颖之处是将图形分解成树形结构及其种子图, 从而能够用相同的树形结构但不同的种子图来识别图形中的尤勒电路。 这种方法也可以用来将平均值和共变系数推至更精确度。