Trace Moments of the Sample Covariance Matrix with Graph-Coloring

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 $\mathit{\text{seed}}\mathit{\text{graphs}}$, 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.