Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

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.

On the Propagation of the Weak Representation Property in Independently Enlarged Filtrations: The General Case

In this paper, we investigate the propagation of the weak representation property (WRP) to an independently enlarged filtration. More precisely, we consider an $${\mathbb {F}}$$ F -semimartingale X possessing the WRP with respect to $${\mathbb {F}}$$ F and an $${\mathbb {H}}$$ H -semimartingale Y possessing the WRP with respect to $${\mathbb {H}}$$ H . Assuming that $${\mathbb {F}}$$ F and $${\mathbb {H}}$$ H are independent, we show that the $${\mathbb {G}}$$ G -semimartingale $$Z=(X,Y)$$ Z = ( X , Y ) has the WRP with respect to $${\mathbb {G}}$$ G , where $${\mathbb {G}}:={\mathbb {F}}\vee {\mathbb {H}}$$ G : = F ∨ H . In our setting, X and Y may have simultaneous jump-times. Furthermore, their jumps may charge the same predictable times. This generalizes all available results about the propagation of the WRP to independently enlarged filtrations.