Random Processes with Discrete Scale Invariance

The Lampertie transform establishes a one to one connection between stationary and self-similar processes. We give a more general connection, in particular between discrete scale invariance (DSI) and cyclostationarity. Some general properties of stationary processes and linear filtering theory, such as linear time invariant systems are reviewed, and their self similar and DSI coun-terparts, such as scale covariant systems are studied. We prove that Lampertie transform maps linear filters to scale covariant systems. We show that any self similar process can be represented as the output of a scale covariant system. Finally we find that the Mellin transform plays the same central role for self-similarity as Fourier transform plays for stationarity.