Canonical Equation K ₃₄ for Normalized Spectral Functions of Empirical Covariance Matrix with Asymptotically Independent Blocks

Abstract

The asymptotic properties of normalized spectral functions of empirical covariance matrices are studied in the case of a nonnormal population. It is shown that the Stieltjes transforms of such functions satisfy the so-called canonical equation. In this chapter, we give only canonical equations. While the proofs of these equations may appear complicated, their main formulas are not extraordinary difficult. The reader, possessing basic understanding of the theory of matrices, can check them without much trouble. This is why, for analogous equations in subsequent chapters, we only indicate the main steps of proofs without giving detailed calculations.