Asymptotic Behavior of Eigenvalues of Variance-Covariance Matrix of a High-Dimensional Heavy-Tailed Lévy Process

In this paper, we study the limiting behavior of eigenvalues of the variance-covariance matrix of a random sample from a multivariate subordinator heavy-tailed Lévy process, and use large deviations of a heavy-tailed stochastic process to derive the limit distributions of its components. We confine our study to multivariate Lévy processes with regularly varying random components and possibly different indices of regularity. Assuming that the product of increments of the marginal components are also regularly varying random variables, we show that the product of two dependent regularly varying Log-Gamma random variables with integer-valued shape parameters is also a regularly varying random variable with index depending on the correlation between the original variables. This result enables us to derive the limiting tail behavior of sample variance-covariance matrix from a multivariate Lévy process having Log-Gamma components with integer-valued shape parameters and different indices of regularity.