Singular properties of high-order spectral densities of supOU processes

This paper investigates the properties of high-order spectral densities and high-order cumulants of superpositions of Ornstein-Uhlenbeck (supOU) processes. As is well known, the long-range dependence of a stochastic process is typically characterized by the second-order spectral density or correlation function. We introduce the property of high-order long-range dependence using high-order spectral densities and cumulants. Sufficient conditions are derived for the supOU process to exhibit high-order long-range dependence. A numerical study of supOU processes with Dickman and gamma marginals is presented to illustrate the theoretical findings.