Correlation cascades of Lévy-driven random processes
ABSTRACT We explore the correlation-structure of a large class of random processes, driven by non-Gaussian Lévy noise sources with possibly infinite variances. Examples of such processes include Lévy motions, Lévy-driven Ornstein–Uhlenbeck motions, Lévy-driven moving-average processes, fractional Lévy motions, and fractional Lévy noises.Based on the fact that non-Gaussian Lévy noises are continuum superpositions of Poisson noises, we unveil an underlying Cascade of ‘Lévy correlation functions’ which characterize the process-distribution and the correlation-structure of the processes under consideration. In the case where the driving Lévy noise sources are ‘fractal’, the resulting cascade admits a unique scale-free form.
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ABSTRACT: In this paper, we investigate the properties of the recently introduced measure of dependence called correlation cascade. We show that the correlation cascade is a promising tool for studying the dependence structure of infinitely divisible processes. We describe the ergodic properties (ergodicity, weak mixing, mixing) of stationary infinitely divisible processes in the language of the correlation cascade and establish its relationship with the codifference. Using the correlation cascade, we investigate the dependence structure of four fractional [alpha]-stable stationary processes. We detect the property of long memory and verify the ergodic properties of the discussed processes.Stochastic Processes and their Applications 119(10):3416-3434. · 1.01 Impact Factor