Local expectations of the population spectral distribution of a high-dimensional covariance matrix

This paper discusses the relationship between the population spectral distribution and the limit of the empirical spectral distribution in high-dimensional situations. When the support of the limiting spectral distribution is split into several intervals, the population one gains a meaningful division, and general functional expectations of each part from the division, referred as local expectations, can be formulated as contour integrals around these intervals. Basing on these knowledge we present consistent estimators of the local expectations and prove a central limit theorem for them. The results are then used to analyze an estimator of the population spectral distribution in recent literature.