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Generalization of the Marčenko-Pastur problem.

Jean-Philippe Bouchaud1, Marc Potters2

  • 1Capital Fund Management & Académie des Sciences, 75007 Paris, France.

Physical Review. E
|January 20, 2021
PubMed
Summary

We analyzed generalized Wishart matrices, finding their Stieltjes transform solves a cubic equation. In a specific limit, eigenvalue density converges to the Wigner semicircle distribution.

Area of Science:

  • Random Matrix Theory
  • High-Dimensional Statistics

Background:

  • Generalized Wishart matrices are crucial in multivariate statistics and signal processing.
  • Understanding their spectral properties is key for analyzing complex data structures.
  • The Marčenko-Pastur law describes the spectral limit for standard Wishart matrices (c=1).

Purpose of the Study:

  • To investigate the spectral distribution of generalized Wishart matrices for arbitrary correlation parameter c.
  • To derive the Stieltjes transform for these matrices.
  • To explore the spectral behavior in the limit of large dimensions (T >> N) and weak correlation (c=0).

Main Methods:

  • Definition of generalized Wishart matrices F = (XYᵀ + YXᵀ)/2T.
  • Analysis of matrices X and Y with zero mean, unit variance i.i.d. entries.

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  • Calculation of the Stieltjes transform and its relation to a cubic equation.
  • Examination of the eigenvalue density in the limit c=0 and T >> N.

    • Main Results:

    • The Stieltjes transform of the generalized Wishart matrix is shown to be a solution of a cubic equation for any c.
    • In the specific limit where c approaches 0 and T is much larger than N (T >> N), the eigenvalue density converges to the Wigner semicircle distribution.
    • This extends the understanding of spectral distributions beyond the standard Marčenko-Pastur case.

    Conclusions:

    • A general analytical solution for the Stieltjes transform of generalized Wishart matrices is established.
    • The study reveals a transition in spectral behavior, linking to the Wigner semicircle in a specific correlation and dimension regime.
    • These findings contribute to the theoretical framework of random matrix theory and its applications in high-dimensional data analysis.