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Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

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  • G. L. Zitelli

    (Capital Group)

Abstract

We prove the existence of joint limiting spectral distributions for families of random sample covariance matrices modeled on fluctuations of discretized Lévy processes. These models were first considered in applications of random matrix theory to financial data, where datasets exhibit both strong multicollinearity and non-normality. When the underlying Lévy process is non-Gaussian, we show that the limiting spectral distributions are distinct from Marčenko–Pastur. In the context of operator-valued free probability, it is shown that the algebras generated by these families are asymptotically free with amalgamation over the diagonal subalgebra. This framework is used to construct operator-valued $$^*$$ ∗ -probability spaces, where the limits of sample covariance matrices play the role of non-commutative Lévy processes whose increments are free with amalgamation.

Suggested Citation

  • G. L. Zitelli, 2022. "Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2176-2193, December.
  • Handle: RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-021-01153-x
    DOI: 10.1007/s10959-021-01153-x
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    References listed on IDEAS

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    1. G. L. Zitelli, 2020. "Random matrix models for datasets with fixed time horizons," Quantitative Finance, Taylor & Francis Journals, vol. 20(5), pages 769-781, May.
    2. Giulio Biroli & Jean-Philippe Bouchaud & Marc Potters, 2007. "The Student ensemble of correlation matrices: eigenvalue spectrum and Kullback-Leibler entropy," Papers 0710.0802, arXiv.org.
    3. Joël Bun & Jean-Philippe Bouchaud & Marc Potters, 2017. "Cleaning large correlation matrices: tools from random matrix theory," Post-Print hal-01491304, HAL.
    4. de Area Leão Pereira, Eder Johnson & da Silva, Marcus Fernandes & Pereira, H.B.B., 2017. "Econophysics: Past and present," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 473(C), pages 251-261.
    5. Silverstein, J. W. & Bai, Z. D., 1995. "On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 175-192, August.
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