Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

G. L. Zitelli ()
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G. L. Zitelli: Capital Group

Journal of Theoretical Probability, 2022, vol. 35, issue 4, 2176-2193

Abstract: 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.

Keywords: Random matrix theory; Free probability; Marčenko–Pastur law; Lévy processes; Infinite divisibility; Primary: 60B20, Secondary: 60G51; Other: 35P20 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01153-x

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