[Paper Review] Spectral densities of Wishart-Levy free stable random matrices: Analytical results and Monte Carlo validation

Mauro Politi, Enrico Scalas+2 more|RePEc: Research Papers in Economics|Mar 9, 2009

Random Matrices and Applications51 references8 citations

TL;DR

This paper derives an analytical expression for the spectral density of Wishart-Lévy free stable random matrices using free probability theory and validates it via extensive Monte Carlo simulations. The key contribution is a closed-form solution for the eigenvalue distribution of covariance matrices built from Lévy-stable distributed increments, showing excellent agreement between theory and simulation across various stability indices and finite-size regimes.

ABSTRACT

Random matrix theory is used to assess the significance of weak correlations and is well established for Gaussian statistics. However, many complex systems, with stock markets as a prominent example, exhibit statistics with power-law tails, that can be modelled with Levy stable distributions. We review comprehensively the derivation of an analytical expression for the spectra of covariance matrices approximated by free Levy stable random variables and validate it by Monte Carlo simulation.

Motivation & Objective

• To derive a closed-form analytical expression for the eigenvalue spectrum of Wishart matrices constructed from free Lévy-stable random variables.
• To validate the analytical prediction using large-scale Monte Carlo simulations of finite-dimensional matrices.
• To extend classical random matrix theory—previously limited to Gaussian or finite-variance distributions—to heavy-tailed, power-law tailed distributions such as Lévy stable laws.
• To demonstrate that the Marčenko-Pastur law generalizes to stable distributions via free probability, even when second moments are infinite.
• To provide a numerically robust and computationally feasible framework for simulating and analyzing such spectra in finite-size systems.

Proposed method

• Derives the spectral density using free probability theory, specifically the R-transform and S-transform of free stable random variables.
• Applies the free convolution of the Marchenko-Pastur law with the Lévy-stable distribution to obtain the eigenvalue spectrum of the Wishart matrix.
• Uses the inverse Stieltjes transform to recover the spectral density from the R-transform of the limiting distribution.
• Implements a Monte Carlo scheme to generate large random matrices with i.i.d. Lévy-stable distributed entries using the characteristic function method.
• Employs random orthogonal rotations to average over multiple realizations and reduce finite-size effects in the eigenvalue histogram.
• Normalizes the covariance matrix using a scale factor derived from the characteristic function of the Lévy distribution to ensure convergence to the theoretical spectrum.

Experimental results

Research questions

• RQ1 What is the analytical form of the eigenvalue spectral density for Wishart matrices built from free Lévy-stable random variables?
• RQ2 How does the spectral density of such matrices differ from the classical Marčenko-Pastur law when the underlying distribution has infinite variance?
• RQ3 To what extent does the theoretical prediction match finite-size Monte Carlo simulations across different stability indices (α) and aspect ratios (m = N/T)?
• RQ4 Can free probability theory be successfully applied to derive the spectrum of Wishart matrices under non-Gaussian, heavy-tailed distributions?
• RQ5 What is the computational scaling of the Monte Carlo simulation method, and how does it depend on the stability index α and matrix size?

Key findings

• The analytical spectral density derived via free probability theory shows excellent agreement with Monte Carlo simulations across all tested values of the stability index α and aspect ratio m.
• For α = 1.5, the theoretical spectrum matches the simulated histogram with high precision, even for finite matrix sizes (N=200, T=600).
• The Monte Carlo simulation runtime scales approximately as T², with the slowest step being the generation of the Lévy-stable matrix Λ, which dominates the computation.
• The method remains efficient for α ≠ 1, with computation times ranging from under 2 minutes (α=1, m=1) to about 47 minutes (α≠1, m=1/6), consistent with theoretical expectations.
• The use of random orthogonal rotations effectively averages out finite-size fluctuations and improves convergence of the empirical eigenvalue distribution to the theoretical one.
• The analytical solution generalizes the Marčenko-Pastur law to the case of infinite-variance Lévy-stable increments, confirming that the semicircle and Marchenko-Pastur laws are special cases of a broader free-probabilistic framework.