Fractal Levy correlation cascades

The correlation structure of a wide class of random processes, driven by stable non-Gaussian Levy noise sources, is explored. Since these noises are of infinite variance, correlations cannot be measured via auto-covariance functions. Exploiting the underlying Poissonian structure of Levy noises, we present a cascade of 'Poissonian correlation functions' which characterize the correlation structure and the process distribution of the processes under consideration. The theory developed is applied to various examples including motions, Ornstein-Uhlenbeck and moving-average processes, and fractional motions and noises-all driven by stable non-Gaussian Levy noises.