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 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.