Eigenvalue Densities of Real and Complex Wishart Correlation Matrices

Wishart correlation matrices are the standard model for the statistical analysis of time series. The ensemble averaged eigenvalue density is of considerable practical and theoretical interest. For complex time series and correlation matrices, the eigenvalue density is known exactly. In the real case, a fundamental mathematical obstacle made it forbiddingly complicated to obtain exact results. We use the supersymmetry method to fully circumvent this problem. We present an exact formula for the eigenvalue density in the real case in terms of twofold integrals and finite sums.

Figure 1

Eigenvalue density for $p=5$ and $n=200$: analytical formula (solid lines) and Monte Carlo simulations (histogram with bin width 3).

Figure 2

Eigenvalue density for $p=10$ and $n=200$: analytical formula (solid lines) and Monte Carlo simulations (histogram with bin width 3).