scholarly journals Convergence Rates of Spectral Distributions of Large Sample Covariance Matrices

2003 ◽  
Vol 25 (1) ◽  
pp. 105-127 ◽  
Author(s):  
Z. D. Bai ◽  
Baiqi Miao ◽  
Jian-feng Yao
Keyword(s):  
Convergence Rates ◽  
Covariance Matrices ◽  
Large Sample ◽  
Sample Covariance Matrices ◽  
Sample Covariance ◽  
Spectral Distributions

scholarly journals Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

2022 ◽  
Author(s):  
G. L. Zitelli
Keyword(s):  
Lévy Processes ◽  
Matrix Theory ◽  
Free Probability ◽  
Levy Processes ◽  
Covariance Matrices ◽  
Sample Covariance Matrices ◽  
Sample Covariance ◽  
Probability Spaces ◽  
Spectral Distributions ◽  
Non Gaussian

AbstractWe 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.

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