Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

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