Asymptotic Coincidence of the Statistics for Degenerate and Non-Degenerate Correlated Real Wishart Ensembles

TL;DR

This paper demonstrates that for large finite matrices, the spectral statistics of correlated real Wishart ensembles approximate those of a doubled, degenerate eigenvalue model, simplifying analysis of complex time series data.

Contribution

It introduces a supersymmetry-based analytical approach to handle singularities in real Wishart models and reveals asymptotic equivalence of spectral statistics between degenerate and non-degenerate cases.

Findings

Spectral statistics of real Wishart matrices approach those of a doubled, degenerate model.

Explicit formulas for the largest eigenvalue distribution in the degenerate case.

Local spectral correlations follow sine and Airy kernels in bulk and edges.

Abstract

The correlated Wishart model provides the standard benchmark when analyzing time series of any kind. Unfortunately, the real case, which is the most relevant one in applications, poses serious challenges for analytical calculations. Often these challenges are due to square root singularities which cannot be handled using common random matrix techniques. We present a new way to tackle this issue. Using supersymmetry, we carry out an anlaytical study which we support by numerical simulations. For large but finite matrix dimensions, we show that statistical properties of the fully correlated real Wishart model generically approach those of a correlated real Wishart model with doubled matrix dimensions and doubly degenerate empirical eigenvalues. This holds for the local and global spectral statistics. With Monte Carlo simulations we show that this is even approximately true for small matrix dimensions. We explicitly investigate the $k$-point correlation function as well as the distribution of the largest eigenvalue for which we find a surprisingly compact formula in the doubly degenerate case. Moreover we show that on the local scale the $k$-point correlation function exhibits the sine and the Airy kernel in the bulk and at the soft edges, respectively. We also address the positions and the fluctuations of the possible outliers in the data.