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 explicit expansion of trace moments and power-trace covariances of $S_{p,n}$ simultaneously for both high- and low-dimensional data. To this end we expand a well-known ansatz of describing trace moments as weighted sums over routes or graphs. The novelty to our approach is an inherent coloring of the examined graphs and a decomposition of graphs into their tree-structure and their \textit{seed graphs}, which allows for some elegant formulas explaining the effect of the tree structures on the number of Euler-tours. The weighted sums over graphs become weighted sums over the possible seed graphs, which in turn are much easier to analyze.

翻译：设 $S_{p,n}$ 表示在零假设下基于 $n$ 个独立同分布 $p$ 维随机向量的样本协方差矩阵。本文的主要结果是同时针对高维和低维数据给出 $S_{p,n}$ 迹矩和幂迹协方差的显式展开。为此，我们将描述迹矩作为路径或图加权和的著名ansatz进行展开。方法的新颖之处在于对所考察的图进行固有着色，并将图分解为其树状结构和\textit{种子图}，这得以用优雅公式解释树状结构对欧拉回路数的影响。原本对图的加权和转化为对可能种子图的加权和，而后者更易于分析。